In a classics lecture hall at the University of Edinburgh an enthusiastic compère bounds on to the stage to welcome the 70 or so fresh-faced students who have gathered for a very different type of seminar. “Well done,” she tells them. “I can guarantee you that just by coming here tonight you will automatically become better in bed.”

They laugh, but while the mood at this hour-long “gameshow”, provocatively titled “How to be good in bed”, is fun, the questions it raises are serious: what does sexual consent look like, what is the difference between flirting and harassment, and how do we tackle a culture of sexual violence?

This term there has been a series of initiatives to make higher education institutions safer, as universities across the UK begin to recognise it is a problem that can no longer be ignored. Guardian investigations have found that sexual harassment at universities is rife, and while some, such as Edinburgh, have grasped the nettle, the response is still patchy.

A report from Revolt Sexual Assault and the Student Room released this year found that 62% of students and graduates had experienced sexual violence including groping, harassment, unwanted touching, coercion and sexual assault, while 8% had said they had been raped.

This week Edinburgh has employed the Consent Collective – a not-for-profit organisation helping communities talk about consent and sex – to carry out a week-long series of events around the thorny subject. As well as three gameshows – which the group estimates have reached about 400 students – they have carried out workshops for survivors of sexual assault and training for staff on dealing with reports of sexual misconduct.

“Most universities realise they have to do something about sexual assault on campus but they are struggling to work out what to do,” says Nina Burrowes, a psychologist and founder of the Consent Collective. “It’s a huge issue and it can’t be dealt with in a piecemeal manner. There’s no way of doing this work without doing all of it. This conversation is about everybody.”

This week is just one part of Edinburgh’s drive to reduce sexual violence on campus, says Gavin Douglas, the deputy secretary of student experience. The university is also working with the city’s other higher education institutions, the NHS, the council and Rape Crisis on a campaign called “fearless Edinburgh”.

Some of the work is common sense, such as improving the university’s website so a search for “sexual violence” signposts users towards help and new guidance for staff rather than research papers, improving reporting and disciplinary procedures.

Others are more innovative. A prominent “No Excuses” campaign has been run by the university’s student association and this year 600 “student leaders”, including captains of sports clubs and societies, have been given training on how to step in to prevent unwelcome or inappropriate behaviour.

“It makes it personal, engaging a percentage of the community – mainly men – about how they conduct themselves [and] how banter is sexist and probably misogynistic,” says Andy Shanks, the director of student wellbeing. “And it gives them strategies to adopt to combat it in their own groups.”

Dave O’Sullivan, the club secretary of the rugby club, did a three-hour workshop this week and is going to do a presentation to other members. “It really got you thinking about how you’d act and getting over that instinct to hold back and instead be guided by your moral compass,” he says. “We have mums, girlfriends, sisters and you wouldn’t want them to be treated like that … if I saw anyone behaving badly, I’d have no problem stepping in.”

Many universities insist they are now tackling sexual violence head-on. Warwick pays for an independent sexual violence adviser from Coventry Rape Crisis to be on campus five days a week and has rolled out campaigns and training on consent; Durham has a full-time sexual violence support officer for students and compulsory sexual violence and misconduct training for all staff and Manchester has launched a bystander campaign to encourage people to intervene if they see something wrong.

Edinburgh saw its number of reported rapes more than double from 34 in 2016-17 to 73 in 2017-18 – a figure Douglas says “ is just the tip of the iceberg”. Other universities doing proactive work have seen negative press reports about high numbers of assaults. These stories make some vice-chancellors nervous. “There is fear among some VCs that the university ends up looking like somewhere where your daughter gets raped,” says Douglas. “You need to be honest about it and people need to understand that the number of reports going up is a positive sign and the first step to reducing sexual violence.”

Douglas credits universities’ response to research carried out by Universities UK and to the National Union of Students for campaigning for more protection for students. But it was attempting to help a student who had been the victim of sexual assault at his own university that made him realise things had to change. “There was a gulf between what she needed and what we could offer her and it just wasn’t good enough,” he says.

One issue that few universities have woken up to is the fact that among their student and staff populations there will be large numbers of people who are already living with the impact of sexual or domestic abuse, says Burrowes. “The #MeToo movement means that more people are being open about their need for additional support at university and at work,” she says. “Universities and employers haven’t been confronted by this yet, but that’s about to change.”

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Sophie*, a 21-year-old art history student and survivor of sexual assault, attended one of the collective’s workshops for survivors and says the week has been genuinely transformational. “I was so scared of going to the workshop but everyone there had an understanding of that pain. You could speak without the barriers; it was OK to cry,” she says.

She has struggled with trauma while at the university and says that when she told one tutor they suggested that if she could not cope she should just take a year out. But after meeting others who have been through similar experiences, she says she feels more able to cope. “I feel so different to how I felt last week, so much more grounded,” she says. “That’s the power of other people. I feel like the barriers are starting to come down.”

*This name has been changed to protect the person’s identity.