Update (Oct. 7, 2011): This bill was signed into law by the governor.
Bill description: Assembly Bill 376 prohibits any person to possess, sell, offer for sale, trade or distribute shark fins. The bill exempts taking sharks for recreational/commercial purposes and persons with licenses/permits to take plant or animal life for scientific or educational purposes.
Read the full text of the bill and follow its progress here.
Note: The text and status of California bills is available here.