But thousands of them are not recognised as refugees by the strict wording of the convention, and many are not given asylum.
Some people say we should rework the convention, acknowledging the different reasons people flee - for example, war, famine, or persecution by non-state agents such as Afghanistan's Taliban or RUF rebels in Sierra Leone.
But others say that receiving countries already cannot cope with huge influxes of people.
Ruud Lubbers, the UN High Commissioner for Refugees, joined us for a special edition of Talking Point ON AIR, the weekly phone-in programme of BBC News Online and BBC World Service.
Select the link below to watch Talking Point On Air