Three major things they have done:
The Pirate Bay Act: In April 2009, after the Pirate Bay co-defendants were found guilty of facilitating extensive copyright infringment, Anonymous responded with DDoS attack against the IFPI.
Megaupload: On January 19, 2012, Megaupload, a website providing file sharing services, was shut down by the US Department of Justice and the Federal Bureau of Investigation. Anonymous called this the single largest internet attack in its history.