Josh Howie jokes about circumcision, the Holocaust and Jewish attitudes to money. Shazia Mirza makes fun of suicide bombers, imams and Islamophobia.
For both comedians, the prime aim is to make audiences laugh. But they are aware that by being funny about such sensitive subjects, they can demonstrate humour's capacity to break down religious and racial barriers .
"It's part of our remit as comics," says Howie, who has been performing on the stand-up circuit since 2002. "Comedy is at its worst when it sets out to offend. But it should prod people. In stand-up comedy, there's a freedom you don't get anywhere else.
"I want my audience to think about the jokes I tell. They laugh, and then should question why it's OK to laugh at something which might seem at first to be racist, to be targeting someone. And then they stop for a second. And hopefully, then there's a second laugh.
Typical of the kind of humour he is talking about is this joke from his old stand-up routine: "I really want to use my comedy to break down negative Jewish stereotypes… because I hear there's a lot of money in that."
Mirza believes that: "A lot of the best comedy is about serious subjects." As long as the material is funny, no area should be off-limits. "I would definitely make fun of, say, the Middle East situation."
But she stresses that changing people's attitudes is not comedy's first priority: "All that matters is that you are making people laugh. I don't sit down and think: 'Well if I write this, maybe it
will make people less racist'."
Racism is an issue both comedians will address at next month's UK Jewish Film Festival's live comedy night, which aims to challenge Muslim and Jewish stereotypes. For Howie, the dynamics between Jews and Muslims is a fascinating topic. "I talk a lot about Jews and Muslims. I'm not taking the mickey out of either of us, but I'm exploring our relationship."
He believes that he is helped by the long tradition of Jewish humour that immediately connects with audiences who are familiar with the work of Woody Allen, Jerry Seinfeld, Larry David et al.
"Being Jewish is like a prism to talk about anything - about Islam, black culture, all sorts of race and sexuality issues, but from a Jewish perspective. That's the starting point."
The situation is different for his Muslim counterpart. Mirza, who has performed gigs wearing a hijab, the traditional headcovering worn by Muslim women, points out that a tradition of Muslim comedy is still taking baby steps.
"We don't have that culture of humour. Jews have been able to laugh through and at their adversity over the years. And we haven't, yet. I hope we learn to do that."
Both agree that self-deprecation is the key to bringing Jewish or Muslim humour to secular audiences.
Howie says: "I always find if I put myself down first then I can talk about these other subjects. If you show that you have a sense of humour about what's important to you, and in my case that's being Jewish, then it gives you permission to talk about whatever you want."
He is no stranger to controversy. His mother is the PR guru Lynne Franks, a one-time Buddhist. He rebelled against his unorthodox background by running away to Israel to train as a rabbi, where he promptly rebelled again and was thrown out of yeshivah after being caught in the company of a non-Jewish girl.
Mirza says that being a female Muslim comedian is a rebellion in itself. Brought up in Birmingham in a family with Pakistani roots - her parents had an arranged marriage - she took a traditional career path, studying biochemistry and qualifying as a science teacher, before succumbing to her ambition to be a comedian.
Eight years into her career, she is moving away from jokes which trade on her background - she no longer opens her set with the line: "Hi, I'm Shazia Mirza - at least that's what it says on my pilot's licence".
"The material has changed a lot," she says. "I have done a lot of comedy about being Muslim. But I don't need to do it as much any more."
Howie's new show, Gran Slam, is also moving away from typical "Jewish" comedy. "This show is a conscious effort to be less Jewish. I'm writing a show about my grandma. Everyone has a grandma.
"Of course, not everyone lives with their grandma for four and a half years like me and my wife did. But the Jewishness still creeps in - she's a Jewish grandma after all."
And when it goes wrong
Comedian Mark Maier discovered what happened when a Muslim joke misfired:
"When I am on stage, I never intend to upset anyone. My most offensive material, according to my wife, is my 1970s blue herringbone shirt. But I wasn't wearing it the night of my performance at The Showcase Cinema, Coventry.
All the more puzzling then, when I received an email, forwarded to me by the Muslim Council of Great Britain, stating I had offended a number of Muslims in that night's audience.
"Not only had I made a joke about the burkha, but I had also 'stuck two fingers up at the prospect of Muslims being offended'.
"I was shocked. I experienced that stomach-churning feeling you get when something is seriously wrong. I was certain I had no recollection of ever saying the word 'burkha' in my act, let alone V-signing the audience.
"The unsettling feeling morphed into anger when it transpired that another act had in fact made the remarks. I was exonorated and received a written apology.
"Four acts performed that night. Why was it presumed I was the guilty party? Could it be because I had the most Jewish-sounding surname on the bill? I don't know, and I would hesitate to suggest such a thing - It could be seen as highly offensive."
Mark Maier's show, 'The Nine O'Clock Shmooze', is at the Arts Depot, London N12, on November 6 and 7. Box Office: 020 8369 5454