Sony has restored most of the PSN services in a rollout that started with North American and European customers before it reached the whole globe.
According to Sony, the PSN services that have been restored include Sign-in for PlayStation Network and Qriocity (including the resetting of passwords), online gameplay for PS3 and PSP, playback of rental video content on PS3 (if within rental period), Music Unlimited powered by Qriocity on PS3 (for current subscribers), access to services such as Netflix, Hulu, Vudu, and MLB.tv, "Friends" category on PS3 (including Friends List, Chat Functionality, Trophy Comparison, etc), in-game leaderboards, and PlayStation Home.
The PlayStation Marketplace and all commerce-related functions haven't been restored yet, but Sony promises to update the Marketplace's content several times per week when it is back online to make up for the missed releases.
The PSN re-launch didn't go as smooth as Sony hoped though. The service has gone offline at least one time after the re-launch and it was noticeably slow several times. Sony has admitted that the large number of users rushing to download the mandatory system update v3.61 which forces them to reset their passwords have put an "extremely heavy load" on their servers and forced them to turn the system off for 30 minutes to clear the queue.
Sony also admitted that the heavy load is causing password reset emails to be delayed for hours, and advised users to be patient.