In the early 1920s the League of Nations was faced with the problem of managing the tremendous number of stateless refugees remaining after World War I and the Russian Revolution. Hundreds of thousands of former Russian, Turkish, Greek, Armenian, and Assyrian citizens were living in refugee camps, roaming the countryside, or living homeless in the streets of foreign cities. These people were often without any sort of identification papers or passport, and what they did have was likely not recognized by the local governments. This made travel difficult, and settling down anywhere in Europe nearly impossible.
The League of Nations formed a Commission for Refugees, and appointed one of their star celebrities, the Norwegian scientist, ambassador, explorer, and humanitarian Fridtjof Nansen, to lead it. Mr. Nansen set down the basics of refugee management as we know them today, although his best know innovation, the 'Nansen passport', is no longer in use.
The Nansen passport was actually as much an idea as a physical object. In practice it consisted of a paper form with basic personal information filled in by the refugee, with a place for a photograph and an official Nansen stamp. However, this was not a standardized from; every state could issue its own form, in its own language, with its own format. In July of 1922, 16 states signed the original agreement; by 1942 a total of 52 countries had agreed to accept (and issue) Nansen passports. This led to a wide array of documents appearing under the label of 'Nansen passport', from single sheets of paper to folded booklets (akin the WHO Yellow Cards of modern day) to over-sized certificates. The local state determined who could receive a passport, how much it cost, how much a visa would cost, and how often it had to be renewed.
Nansen was careful not to step on any toes when setting up this process, and to this end there were some limitations to the Nansen passport. It did not give all of the legal rights of a passport; most importantly, it did not give the bearer the right of return, that is, when a country issued a Nansen passport it did not issue the right to return back to that country once you left. It was also made clear that the Nansen passport was not intended to change or impinge on any laws of the issuing countries. While all of the governments that signed the agreement promised not to expel refugees as long as they were not a threat to national security or to public order, the interpretation of this promise was largely up to member countries, and whatever laws those countries may have had in regards to foreigners and refugees remained in effect.
The somewhat chaotic and laissez-faire system of administration and legislation led to some problems. In Russia they were called 'nonsense passports', as they were nearly useless to many holders. For example, in France it remained the law that any foreigner convicted of any crime was to be deported. However, a Russian refugee was officially stateless, and was simply escorted to the border and told not to come back. If this visa-less individual was caught by border guards of the neighboring country, they were sent back -- to France. There was no legal recourse provided to help one escape this potentially never-ending cycle. Any number of nonsensical catch-22s could lay in wait for the hapless refugee.
However, the passports were far from useless. The passports were primarily grease for the wheels of bureaucracy. No longer were refugees automatically turned away from the border simply because they had no papers. Now when a refugee tried to apply for a real passport, transportation to their home country, or citizenship in their new locality, they had a starting point, something to show the man behind the desk. Despite the Russian scorn for the passports, Igor Stravinsky, Sergei Rachmaninov, Marc Chagall and Anna Pavlova, among many others, owe the happy endings to their expatriations to Nansen passports. Had this paperwork not been available, they might have remained as outcasts and second-class citizens for the rest of their lives.
I can find no accurate estimate of the number of Nansen Passports issued; numbers posted (mostly as mirrors of Wikipedia) confound the number of passports with the number of refugees repatriated during Nansen's first few years working with the League of Nations. The number is certainly in the hundreds of thousands.
Nansen passports, as such, are no longer issued, but stateless refugees may still be issued documents by various governments and organizations. These documents are now known by a host of labels, including 'Certificate of Identity', 'Alien's Passport', 'Refugee Travel Document', 'Emergency Passport', and Laissez-Passer.
Nansen Passport, Dead Media Archive
Russians outside Russia: the émigré community in Czechoslovakia 1918-1938 By Elena Chinyaeva.
The United Nations Office at Geneva: About UNOG
Nobel Prize: Fridtjof Nansen
Nobel Prize: The Humanitarian Nobel Peace Prizes
Wikipedia: Travel Document
More pictures of Nansen passports