The current Mulligan system, the Paris Mulligan, was developed as a replacement for the original mulligan system. It was meant to be used at Pro Tour Los Angeles, but by mistake the floor rules packet for Pro Tour Paris included the description of the new mulligan rule as well, hence the name Paris Mulligan.
To mulligan, after drawing hands but before the game starts, the players decide in turn order if they would like to mulligan or keep. Each player who chooses to mulligan shuffles his or her hand into their library and draws a new hand with one less card than before. Each player who chooses to keep has his or her hand become their starting hand and is no longer able to take mulligans that game.
A popular variant on the Paris Mulligan is used in the Commander format, called the Partial Paris, wherein instead of putting their entire hand back into their library, a player may choose to put a number of cards from their hand aside face down, then draw that number of cards minus one, after which they shuffle the cards they set aside into their library.