Three years ago, I met my fiance. He was fresh out of a 15-year relationship and concluded that a single partner was not what either of us needed. Though I cried when he told me this, I could just about envision a committed scenario without monogamy. So that's what we did.

Like most open couples, we began with dozens of rules: who should call who when, what partners would be OK. But it quickly became clear that these attempts at control were aimed at avoiding jealousy, and that most negative feelings were not jealousy at all: they were my own fears – that he would leave me, or that I wasn't the epitome of sexuality in his eyes. I grew to understand that though he loves my body, he's also sometimes attracted to other bodies; that one day he might leave me, and refusing him access to other sexual experiences won't change that.

As I became more secure in the relationship, the rules faded away, leaving just one: no surprises, which means pre-scheduled dates and no sudden, "I just slept with Susie!" announcements. Though sex is always hypothetically on the table, it's not really: after years of navigating the finer tones of friends versus lovers, I am pretty clear whether my connection with someone is sexual- and/or love-driven, and proceed accordingly.

Arianne Cohen, who has been in an open relationship for three years. Photograph: Dan Tuffs for the Guardian

On my last date, a friend who knows my fiance came over to change my car headlight. We had some wine, talked about his recent break-up, and ended up in bed. I reported back to my partner, as always – our rule is full disclosure when asked; he usually asks more than I do. That same week, he spent a sunny day roaming the city with a woman he's been seeing. I was at a workshop, and happy he had something to do. It's normal, like going to the cinema or calling a friend. I find it largely unremarkable; my friends have long since lost interest.

We rarely see each other's partners; some people do it differently. Claire, a small business owner and amateur musician, and Bill, a technology consultant from Oxford, frequently socialise together with their lovers. They are in their mid-40s and have been together for 24 years. She has a boyfriend, Chris, of seven years; Bill has a girlfriend, Julie, of eight years, who is in a long-term relationship with her partner George. "From an emotional point of view, it's been pretty straightforward for the last many, many years," Claire says. Bill and Chris sometimes attend Claire's performances: "People probably wonder why I keep turning up to my gigs with two blokes. They've never said anything, naturally."

The relationship works so well that Claire struggles to think of recent friction. "Two years ago, there was a moment when Bill ran up and said, 'Julie's pregnant.' And I said, 'By George, right?' And he said, 'Yes.' That was the right answer." Claire last saw Chris on Tuesday, while Bill was rock climbing. "I cooked him some dinner and we caught up on our weeks. We are in contact during the week but not every day. We had a couple of drinks and ended in bed." She is Chris's only partner. "He mentioned something about snogging at a party a few months ago, but I think that's it. He likes his own space." Bill last saw Julie two weeks ago. "I went to her place after work, and waited for George to get back from work. We handed over custody of the child, went out for dinner, had a nice meal of sushi, came back. I waited while Julie performed her breastfeeding duties, went to bed, managed to stay awake to have a bit of sex, then collapsed into torpor. George was in the house looking after the baby." Both George and Julie have other lovers, and an extra bedroom devoted to the purpose; as far as the child will be concerned, these are just Mum and Dad's good friends.

The assumption that Bill or Claire would be racked by jealousy is called mono-normative thinking – an assumption made by monogamists. "There seems to be no logical reason why you couldn't have more than one relationship, so long as both parties were happy," Bill says. "But again, I've only ever dated two people. And I'm still dating both of them."

What is most intriguing, though, is that despite Claire's laid-back attitude, she keeps her relationship choices a secret. "My family – we're pretty private people in that regard. It's not their business. They have met our partners socially, but not had them introduced as such. I devoutly hope my parents know nothing at all."

This seems to be a particularly British take on non-monogamy: comfort with the act, mixed with a compulsive need for privacy. Doing it? Fine. Speaking about it? Never. Claire struggles to articulate this side of her life. "It doesn't come up terribly often, because most of the people to whom I might be describing it already know. I guess if I was describing it, I'd say 'open'. Of our larger friend group of 25, it's not new. Maybe a third are currently in open relationships." I asked Claire if anyone had ever reacted badly. "Not recently, and possibly not ever," she says. Her advice for other potential non-monogamists is straightforward: "Think first. Discuss first. Don't be an idiot. Rules of life, really."

Non-monogamous relationships are surprisingly common and the numbers are increasing, according to Darren Langdridge, a clinical therapist, professor at the Open University and co-author with Meg Barker of Understanding Non-Monogamies.

But statistics are hard to come by, Barker says. "No national surveys cover anything like open relationships – and many people are not upfront about being in one." Through the 60s and 70s, communes, swinging and group marriages entered the mainstream, and then faded when HIV emerged. "Suddenly it was not acceptable to talk about open relationships," Langdridge says, "even though the fear was a myth – you can have as much sex as you'd like with others, and it's safe, as long as you're safe." Which means condoms, 100% of the time.

From 1980 until around 2005, most relationship researchers in academia and public health couldn't get funding for their out-of-vogue topic. Media coverage disappeared. Which is why you might be the unknowing London neighbour of Rekha, 32, who works in publishing. Rekha has been with her boyfriend, who is a doctoral student, for eight years. They're emotionally exclusive, but not sexually. "We've publicly expressed to our family and friends that we intend to spend the rest of our lives together. At the same time, we have different kinds of intimate relationships with other people of both genders: it can be casual dating, or right now, I have an ongoing relationship of about a year."

The man Rekha has been seeing is not someone she would date monogamously. "He's not really a partner, more of a close friend I'm attracted to. We have a bit of age difference, and lifestyle difference. As friends those things don't get in the way, but we probably wouldn't work as a couple. Some of the point of being in an open relationship is finding an arrangement that fits your own needs. At times we've said, 'Let's cool it for six months, we have a lot of stress with work and need to be there 100% for each other.'"

Her family has no idea. "It's not something we discuss, but that has more to do with not speaking with family members about sex. I'm quite clear with my family about not having a completely couple-centric view of the world – that's partially why we had a commitment ceremony rather than a marriage."

A large minority of non-monogamous adults are midlife divorcees who, after long-term monogamy, are keen to try something else. Max, 48, emerged in her early 40s from a 20-year relationship and two sons. "I was married to a lovely guy, but he was so controlling, even about going out with a girlfriend," she says. "When I met Richard" – at a party – "he was the most straightforward man I'd come across. He was, like, 'This is what I do.' We went on a few dates, and Richard said, 'We get along so well, and you'll never stay over because your stuff isn't here, so why don't you move in?'" And so she did, two years ago.

Max with her partner, Richard. Photograph: Ellen Nolan for the Guardian

Richard, 37, summarises the open part of their relationship as "going to sexy parties together, and going out on dates with other people from time to time". The pair practise non-monogamy differently. "Richard is much more actively seeking sexual interactions than I am," Max says. "I want him to have whatever he wants, and I think what he wants is frequent interactions with lots of different women, rather than three regulars. He doesn't have an ongoing relationship right now." Max, meanwhile, has what she calls her shortlist. "There are three active people on the shortlist right now. If I'm feeling like I need a little attention or need a diversion, they're just friends I sleep with. Saying 'lovers' makes me feel like Simone de Beauvoir."

Scheduling is handled digitally. "The idea," Richard says, "is to give sufficient time for Max to do something else, so she's not knocking around unless she wants to be."

"I don't date very often," Max says. "I like snuggling up at home. It's probably two or three dates per month. And maybe once a month I actually sleep with someone else."

"My last date," Richard announces, "ended somewhat disappointingly, because I was not sufficiently direct in my communication. I invited someone for dinner, and she came around for dinner. That wasn't what I meant. Lesson learned."

Max is happy to leave the parties to Richard. "I tried it. And it was all right. I have a homing beacon that comes on at 2am. So I come back and go to sleep and leave him. And it's great. I'll get the bed to myself." She doesn't ask for details. "Basically, all I want to know is, did you shag her? No details necessary. It's something I'm still working on."

Richard has, a few times, become infatuated with women. "I have a tendency to idealise strangers, and then as soon as I've got them, I lose interest. So for me, having this open relationship means I can move through that pattern. And because it's out in the open, I can talk about it, and Max can take it less personally. It loses its power."

Both have had to contend with unpleasant emotions. "Richard goes out with a girl who's 25, and I need to deal with it. She doesn't have wrinkles or baggage or life experience. It forces me to grow. I have to know who I am. And be solid and happy in an open relationship." Richard has not dodged this bullet either. "There's a 26-year-old cop on my shortlist," Max says. "If I go and sleep at the policeman's house, it makes Richard a little grumpy. And I'll help him. It's OK – I don't want to live with a policeman. He's just a friend."

What does Richard wish that others would understand about his relationship? "People think monogamy rescues people from the terror of attraction and abandonment. But it doesn't."

"I think," Max says, "that people look at us and see whatever they are afraid of. So they say, 'You must feel so jealous.' They don't understand how close Richard and I are. You have to be honest. And brave."

Lori Smith, 36, a university administrator, did not set out to be in an open relationship. "We were both monogamists," she says of Jon, her partner of 13 years. "But in the first year, Jon was tempted by someone else, and we realised that it didn't affect how we felt about each other." They talked it over extensively and decided to explore where that could lead. Jon Googled "swingers Manchester", which began a five-year habit of monthly parties. By 2006, Lori found that the parties were "wearing thin, not quite as exciting. Jon was spending time with a woman he'd met on Facebook. And it was fine, just sex" – but very different from their swinging agreements. "We thought, well, what's the difference if we wanted to go further and have a romantic relationship instead? We had big long chats about how we'd feel. We wanted to have the discussion beforehand, not when one of us came home and said, 'I've fallen in love.'"

Lori decided to begin calling herself "polyamorous" – a term that means pursuing multiple consensual love/romantic relationships. It's a subset of non-monogamy, the blanket term for more than one sexual partner. The term polyamory is only 20 years old, and has entered the lexicon because it emphasises love: it's much more socially acceptable to talk publicly about multiple loving relationships than multiple fuckbuddies. The polyamory movement is driven by grass-roots activists – around 200 people appeared at London's PolyDay last August. In the US, polyamory has a hip connotation, and suffers from an epidemic of promiscuous people hiding behind the word. In the UK, polyamorists tend to be more hidden. "What I see in the [UK] movement is it's the radical fringe – people with pink hair and tattoos," says Deborah Taj Anapol, a clinical psychologist and author of Polyamory In The 21st Century. "These are people who don't mind being judged or excluded from mainstream society – in fact, that's their intent. That's all fine, but I'd like to see a quicker normalisation." Which is why many non-monogamous Britons won't use the word. "It seems to be a loaded term," Lori says. "For a while we said non-monogamous, but now we just say 'poly'."

Once Jon and Lori decided to be polyamorous, Jon joined the free dating site OkCupid, known for its large, non-monogamous contingent, and began enjoying weekend dates with a woman who lived just outside London. Lori dated a photography classmate, but struggled more than Jon with the situation. "Once a month Jon's girlfriend came around, or we'd all go out for dinner. And we got on fine, but I just felt really uneasy when they were spending time without me. I couldn't wrap my head around it, so I saw a therapist." Lori realised she suspected that the woman would hurt Jon. "I realised that I needed to let go, let him explore this for himself."

Jon now has a different girlfriend of a year, Amanda. Lori asked to meet her a few weeks in – and discovered that they got on well. "We meet for coffee or lunch a few times a week. At the start, we both thought it was a bit strange, but now not at all. We love each other, but not in a romantic or sexual way. We're best friends." Jon spends Wednesday nights at Amanda's house; she visits most weekends. They have all had sex together, but not often. "When all three of us sleep in the same bed, it's just sleeping. It's nice to spend time together and not have to have sex be part of it."

Some names have been changed.

• This article was amended on 23 October 2015 to remove personal details at the request of the individuals concerned.

