The basis of the Museum of London's new exhibition Belonging is hundreds of hours of interviews with London's refugees, describing their lives from first arriving in Britain, to the new existences they have built over ten, twenty or fifty years. The picture that emerges is very far removed from the image of benefit-grabbing migrants of the tabloid press.

The museum has deliberately embraced this edgiest of political subjects, and clearly wants to start a debate. Belonging lays out the testimony of many who have made the transition from asylum seekers to citizens - whilst inviting museum visitors to take the Citizenship test and consider the long process of getting leave to remain.

The entrance to the exhibition is a bland corridor - similar to the ones used to walk off a plane. It is filled with the voices of refugees telling snippets of their stories. Inside, the exhibits challenge the norms of museum objects. Often arriving in this country with almost nothing, new arrivals invest a blanket, a toy, a set of tea cups with huge significance. So on display we see the cotton blanket that Alemu T Ayele brought from Ethiopia and slept under during his first nights in Britain. We also see the micro economy that operates amongst refugees. Hooshang Paigir, from Afghanistan recalls the first three years he spent in London, living in a hotel:

when I first came here, I had a small pan, small pot for tea, some cutlery, which could all fit in a small box. When I built up my life and I bought my own cutlery and kitchen materials, I kept my first kitchen stuff and used to lend it to new comers in the hotel. I used to tell them that I will lend you these [laughs], but you have to return them back to me. When they finished with them, I used to collect them. When I got the cutlery back, I used to clean it nicely and whenever there was a new comer, he or she would have benefited from it. Like that I used to help them."

His knife, fork and plate are neatly on display.

But these are not stories that continue as records of scraping a living at the margins of society. What's consistent is the level of entrepreneurship and reinvention that's been necessary for all. 23% of asylum seekers are graduates, but often their experience becomes worthless when they arrive in the UK. Mahdi Mahdi, a doctor from Iraq, was unable to retrain in London and therefore began a small food business. Now he has two factories and employs 30 people. Haile Mercurios' story has an edge of comedy:

"It is difficult to explain at the job centre. I told them I used to be a salesman, I had to say that, otherwise they would not take me to anything if I said I was a General"

By contrast to the blankets and plates, there's a painting of Dr John Sentamu arrayed in splendour as he is enthroned as the new Archbishop of York. Dr Sentamu was another refugee obliged to completely change career on coming to Britain - originally a magistrate in Uganda, he fled Idi Amin's regime in 1974.

Despite the subject matter, this is a story about and for children too. Storybooks and drawing activities sit at the edges of the exhibition. Marek Tanoos described his school experience in the Czech Republic in his application for asylum made in 1998.

"Children of the school used to call me names and if I retaliated by pushing them I was punished, and had to sit out in the corridor at school.... when I was eight years old the school tried to get me transferred to a Special School but my parents refused to let me be sent. It was very common for Romany children to be sent to special schools and this can blight the rest of our lives. If you try to get a job when you leave school and you tell an employer you have been to a Special School the chances of you getting a job are nil."

His words are disturbingly reminiscent of oral histories of schoolchildren from sixty years before in the Holocaust Galleries of the Imperial War Museum. These also describe sanctioned bullying at school, but as the first step to a much darker outcome.

Indeed, the Museum of London counterpoints its modern oral histories with some of the oldest recordings in its collections - interviews with Eastern European Jews who came to this country from the 1890s to the 1940s. Among these voices is Ruth Tosek, who came to Britain as a refugee in 1939. She describes returning to the Continent and miraculously finding her skeletal mother still alive in the newly liberated Belsen concentration camp.

It was in the aftermath of these terrible events that the Universal Declaration of Human Rights was drafted, containing the words "Everyone has the right to seek and to enjoy in other countries asylum from persecution".

Belonging argues powerfully that this right is not a question of imposing a burden of scroungers on the host country. The people who speak in this exhibition show great courage and determination to adapt and contribute.

As you wander amongst the 'pavilions' of the exhibition, pick up headphones and listen in you get that rarest of things in the refugee debate - the plainly told stories of the people themselves. More than fifty years after the Declaration of Human Rights, an exhibition like this should not be groundbreaking, yet it is.

The Museum of London's Belonging exhibition continues until 25th February 2007, admission free. There are many tours, performances and events associated with the exhibition - further details here