According to the BBC, Snowden has been issued a temporary document by the Russian Federal Migration Service while they are considering his asylum application, and he will be free to officially enter Russian territory.
The document will be delivered to him by Anatoly Kucherena, his Russian lawyer, and he is expected to leave the airport's premises soon after. His travel within Russia will not be restricted, and he is free to settle anywhere he wants.
Snowden has been offered asylum by Bolivia, Venezuela and Nicaragua, but he has to have a valid traveling document to reach any of those countries - that is, of course, if he accepts any of these offers.
According to Kucherena, if Snowden's asylum request is granted, he intends to make Russia his home for the time being.
"He’s planning to arrange his life here. He plans to get a job. And, I think, that all his further decisions will be made considering the situation he found himself in,” the lawyer stated for RT.
It can take up to three months for the Federal Migration Service to make a decision, and if asylum is given, it will be temporary and Snowden will have to repeat the request every year until he decides to permanently leave the country.
What this move means for US-Russia relations and the potential leaking of the other documents Snowden managed to exfiltrate before his run remains to be seen.
Russian president Putin said that Snowden will be allowed to stay in Russia if he agrees to stop leaking confidential NSA information, but according to The Guardian's Glenn Greenwald, that decision is now out of his hands as he has already shared all the documents he had with journalists.
UPDATE:Newest reports seem to contradict earlier ones, and Snowden is apparently still stuck at the airport, as his lawyer has brought him documents and books but not the pass that would allow him to enter Russia.
According to USA Today, Kucherena has reiterated Snowden's intention of staying in Russia and says that he is using his extensive free time for learning the language and culture. It is unknown when the aforementioned pass could finally be issued.