One day last summer, Inna Shevchenko went into a forest outside Kiev, to learn how to use a chainsaw. The lumberjacks who were instructing her couldn't work out why she was so keen. "They thought I was just a crazy blonde," she says, shaking her white curls. "I was acting like: 'Oh really?'" She affects a coy, clueless demeanour. "'That's how you do it? Great!'"

The next day she went to a hilltop overlooking Kiev, and stripped to a pair of red denim shorts, worn with heavy boots, leather gloves, and a mask to protect her eyes. The Pussy Riot verdict was due that day, and in tribute to the Russian punk activists – and to mark her opposition to all religions – Inna proceeded to chop down a 13ft wooden cross that had been there since 2005. Despite her preparations, it wasn't easy. "When I started to cut it, I thought, 'it's not possible to destroy it,'" she says. But after seven minutes it fell, and she posed against the stump for invited journalists. With "Free Riot" scrawled across her bare breasts, she held out her arms to mirror the figure of Christ now lying on the ground.

Femen protester Inna Shevchenko cuts down a 13ft wooden cross overlooking Kiev, in tribute to Russian punk activists Pussy Riot. Photograph: Sergey Dolzhenko/EPA

Death threats arrived instantly. She says there were official calls for her arrest, and Russian TV reported that the cross was a memorial to the victims of Stalinism. Inna denies this, but Ukrainian journalists repeated the claim, and anger towards her sharpened. Men she suspected of being secret service agents immediately began milling outside her apartment, and a few days later, she was woken at 6am by the sound of her front door being kicked in. She escaped, jumping through a back window, then down from a first floor balcony, and made her way to Warsaw with $50, a mobile phone and her passport. She feared jail if she returned to Kiev, so some days later, she travelled to France, where women had expressed interest in joining Femen, the feminist group she runs with three Ukrainian friends.

Femen's aims are straightforward, broad and radical. A war on patriarchy on three fronts, calling for an end to all religions, dictatorships and the sex industry. The group has been offered a space in a rundown theatre in Paris as headquarters, and it is here I meet Inna, 24, at the start of a training session with 20 young Femen activists. She is giving instructions on the correct stance – feet apart, firmly rooted, aggressive. Femen warriors never smile, she says, they are not there to please anyone. The group has been protesting topless since 2010, using their bodies to attract attention, to lure journalists, and they have been roundly criticised by some people, who accuse them of playing into sexist stereotypes.

In a room covered with posters and murals – Fuck Religion, says one, Go out! Undress! Win!, says another – the activists stand in rows, screaming slogans at each other. They're dressed in T-shirts and tracksuits, occasionally stopping to swig bottled water. This is gym class for the revolution.

"Not a sex toy," they scream. Then "Poor because of you" and "In gay we trust". One by one, they take to the middle of the room, to show how they would behave at a protest. One new member shouts "Pope No More", before two other activists launch themselves at her. For a moment all three are mid-air, then they hit the ground and start struggling in a blur. Inna has told them they must move constantly, to avoid being covered; their slogan will be written across their bare chest and back, and they need it to be seen. One woman fights hard, still screaming, occasionally breaking free, running a few paces, only to be brought down again with a brutal thwack. Finally, Inna calls a halt, and the woman stands up with blood running down her arm. Inna smiles, grabs her hand, and holds the injured limb aloft. There is clapping, cheering, congratulations.

As the activists start the next stage of training – situps, press-ups, running-while-screaming – journalists and cameramen swirl around. There is no attempt to hide the fact this session is being played out for the press. As women fight, Inna comes up close to them, in her denim hotpants, hooded top and Converse boots, instructing them to look at the camera. It doesn't matter how many people come to a protest, she says – if there's one camera, that's what they need to target, to get their message out to millions.

On some level, this is working. Each time Femen stages an action, videos pop up on websites worldwide. But are their breasts obscuring their message? When I tell a friend I'm due to interview them, he is fascinated by the idea of topless feminist warriors – but switches off as soon as I mention their arguments. I suspect there are long-time feminist activists who take one glance at their tactics and, jaded by the use of women's bodies in art, advertising, commerce, don't pause to hear what they're saying.

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Their message can also get lost in the breadth and sprawl of their protests. While other groups focus on one or two issues, Femen are everywhere. Over the past few years they have protested for gay rights in St Peter's Square during the Pope's weekly prayers; against the use of ultra-thin models at Milan fashion week; and during Euro 2012, in Ukraine, they grabbed the championship trophy in protest against the sex industry. In London last summer, they smeared themselves in fake blood and accused the International Olympic Committee of supporting "bloody Islamist regimes"; at Davos, in January, they protested against male domination of the world economy. And in February, they provoked both raised eyebrows and a few sniggers by launching themselves topless at Silvio Berlusconi.

Their campaigning is unified by one central aim: to use their breasts to expose corruption and inequality wherever they see it. "One of the main goals," says Inna, "is to take the masks off people who wear them, to show who they are, and the level of fucking patriarchy in this world, you know?" She says they also want to reclaim women's bodies for women. "A woman's naked body has always been the instrument of the patriarchy," she says, "they use it in the sex industry, the fashion industry, advertising, always in men's hands. We realised the key was to give the naked body back to its rightful owner, to women, and give a new interpretation of nudity ... I'm proud of the fact that today naked women are not just posing on the cover of Playboy, but can be at an action, angry, and can irritate people."

The group started in 2006, when founders Anna Hutsol, Oksana Shachko and Alexandra Shevchenko (no relation to Inna), became friends in their home town in Ukraine. It was not long after the orange revolution, in which Ukrainians had demonstrated for democracy, and Alexandra, 24, says they wanted to keep the revolutionary feeling going. They started a women's group, and began organising against the sex industry. Sex tourism is a major problem in Ukraine, and every woman is victimised as a result, says Alexandra. "You'd walk down the street and foreigners, men, would come up to you, ask how much, touch you."

Inna joined the group in 2009, after meeting the other women on social media. In those early days they were just developing their views. Feminism was unpopular in Ukraine; saying you were a feminist was "something similar to saying you're an idiot, you're crazy," says Inna. Alexandra says she used to believe the "image created by patriarchy, where feminists are ugly women with moustaches who want to cut off men's penises". (They've played with this imagery themselves. Until recently, their website featured a picture of a woman holding an enormous scythe in one hand, a bloody scrotum in the other.)

They embarked on long, lively discussions about women's rights. "We're not based on 700 pages of doctrine," says Inna, "instead we would come in and saying 'can you believe that fucker? He just touched my arse and said he wants to fuck me, and he will pay me with a cocktail.' The discussion was very primitive, and we became angry, and wanted to express it, so we started doing street performances."

These were clothed at first. They would go out with price tags hanging off them to protest against the sex industry, for example; the group has long called for the Swedish approach to prostitution, in which clients rather than sex workers are criminalised. They always had an interest in branding, and initially wore all-pink outfits – some journalists called them the pink revolution. In 2010, in protest against the appointment of an all-male cabinet, they dressed up as men, then took off their suits to reveal women's clothing. Inna was working in the press office of the Kiev mayor at the time. As a result of the protest she was fired.

Femen activists protest outside Notre Dame in celebration of the resignation of Pope Benedict and the French parliament's decision to approve a draft law allowing same-sex marriage. Photograph: Joel Saget/AFP

That same year, they staged their first topless protest. Five activists at the polling station where presidential candidate Viktor Yanukovych was due to vote stripped in front of the media. The next day, the image went global. They weren't immediately convinced they should protest topless all the time, but the response started "a huge, hot aggressive discussion," says Inna. She was initially against the idea. She felt shy, uncomfortable with her body, and frustrated that as "a woman you have to take off your shirt to say something". But they continued to protest – sometimes topless, sometimes clothed – for six months, and she "realised that sometimes when we were not topless, we were talking about much more important topics, but they were just ignored". They decided to go for it, and grew wise to the tabloid tactic of cropping out their banners – they began scrawling slogans directly on their breasts. Inna says her naked body now feels like a uniform, while Alexandra describes it as, "my weapon, my gun".

The group has been accused of deploying only young, slim, beautiful women. But a new book about Femen, just published in France, features photos of women of different shapes and sizes on demonstrations, pictures I've never seen elsewhere. The media, unsurprisingly, pick the most obviously attractive photos. Inna says they have never chosen women according to their looks, or weight; the only proviso is that they have to be well-prepared. "There are a lot of girls who are very strong physically, but they cannot show aggression, they cannot imagine how they will react if someone tries to grab them." The movement is non-violent – Inna calls it "peaceful terrorism" – but she has been injured more than once, and was badly beaten during a recent action.

Their actions have sometimes been dismissed but there is no doubt the women of Femen take serious risks. In late 2011, for example, Inna and two other activists travelled to Minsk, in Belarus, to protest outside the KGB offices against Alexander Lukashenko, the man often called Europe's last dictator. While they expected to spend New Year in a Belarus jail, they allege that they were, instead, abducted by secret service agents – a claim the Belarusian KGB denies.

Inna says a group of men caught up with them at the bus station, and they were driven for five hours into a forest. There, she claims, they were covered in oil, threatened with lighters and knives, and ordered to strip completely. She assumed they were going to be raped. "They put handcuffs on us, and we were sitting like this," she leans forward with her hands behind her back, "for six or seven hours, not allowed to move or talk. One guy kept repeating that we were going to be killed, but before that they were going to have fun.

"They said, breathe, enjoy the air, because there is only one hour left of your life. Imagine the smile of your mother, and now imagine her face when she gets your dead body." Inna felt sure they would be killed, and started analysing her years with Femen. "I knew this was the best part of my life, and something I would never feel sorry about, even in a situation when I could be killed, and it was the greatest answer for me. It was one of the worst days of my life, but also the best, because I understood myself." The three women were eventually dumped in the snow, and she says the incident made her more determined than ever. "I suddenly saw the huge potential of this. Maybe it's strange to say this – I know some people already think we're kamikaze – but that's why I now say I'm more of an activist than a person, because I know that tomorrow I could be killed."

Alexandra says there is a criminal case against her at the moment in Ukraine, where she has been imprisoned a few times for her protests, but she is living in Berlin, organising the activists of Femen Germany. The group wants the movement to spread globally, and they try to support women who start offshoots in their own countries. They now have about 200 activists worldwide – a small number, but able to make a major impact – with branches in Switzerland, Poland, Holland, Sweden, Brazil, USA, Canada and Italy. An activist in Tunisia recently posted a topless image of herself online, and two days ago it was reported that a fatwa had been issued, calling for her to be stoned to death.

Apparently there are a few UK activists, and one British woman, Pippa, 25, who lives in Berlin, has been protesting with the German group. She says she appreciates how active they are. She was involved in student feminism in the UK, but found it grindingly difficult to get people interested in protests. When they started Femen, says Alexandra, they felt they needed to change the way feminism was communicated to young women. "They don't want to read huge texts," she says, so the key was to create something visual. "We understood that people have a lot of information coming at them through mass media, and we needed something that could shock people, shake them, grab their attention."

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Pippa likes the fact they don't spend hours debating actions; they just get out and do them. But this approach might cause them trouble in future. In Germany, for instance, they've been criticised for comparing the sex industry to fascism, using Nazi imagery to underline this comparison. There seems little doubt they hate fascism – they protest regularly against extreme rightwing groups, who on one occasion knocked Inna's teeth out – but Alexandra is determined they should continue to use this loaded metaphor, despite protests from Femen Germany activists. "I understand that they feel this pain," she says, "but we want to make this connection between prostitution and fascism, because people know that fascism is a fucking bad thing." Inna uses the comparison in the context of religion too. "I strongly believe that one day religion has to be forbidden," she says, "the same way fascism was forbidden."

Femen aren't subtle, they aren't inoffensive, and they certainly aren't sorry. "We're provocateurs," says Inna, "and the reaction depends on those who are provoked." With members having faced loss of livelihood, alleged abduction, arrest, jail, death threats and ridicule, it seems they are in it for keeps. "One of our slogans is: 'Fight until the last drop of blood,'" says Alexandra, while Inna notes that every morning she wakes up to death threats, sent via text message, that simply say "die", or "burn". When she sees them she thinks: "Good morning!" she says, laughing long and hard.