Police raids on the offices and home of Kim Dotcom led to the closure of Megaupload.
The Mega site went online on Sunday, followed by a lavish launch party held at Mr Dotcom's New Zealand mansion.
Mr Dotcom, who was born Kim Schmitz, has said the new site complies with the law and warned that attempts to take it down would be futile.
"This is not some kind of finger to the US government or to Hollywood," he said on Saturday.
"Legally, there's just nothing there that could be used to shut us down. This site is just as legitimate and has the right to exist as Dropbox, Boxnet and other competitors."Extradition hearing
Hours after the site was launched, Mr Dotcom tweeted that it had received 250,000 user registrations, although limited server capacity meant Mega was unreachable to many.
In a series of earlier tweets Mr Dotcom said every customer would have 50 gigabytes of free storage - far more than is offered by rival services such as Dropbox or Microsoft's SkyDrive.
Data is also being held in the cloud to make it easy for users to get and share files.Mega will be encrypted so only those who upload data have access to it.
