By the time you read this the PlayStation Network should be back online in most regions in the United States. After three weeks of being offline, the service slowly started going back online after Sony issued a new firmware update to deal with firming up security and allowing users to change their passwords.
The first phase of the network restoration will include online play, video streaming, friends lists, PlayStation Home, and Qriocity. Unfortunately the PlayStation Store is still offline. The v3.61 is mandatory and is available now.
And on a related note Sony Online Entertainment's various online game services are coming back online this weekend as well. From the SOE Site:
"We are pleased to be back online and deeply apologize for the inconvenience caused by the recent data breach and subsequent outage.
To thank you all for your patience and understanding, we have prepared a Welcome Back Program that includes complimentary identity theft protection, free game time and special in-game items and events.
Reminder: As an added security measure, we require a password reset for your Station Account - we recommend you do this immediately."