A time machine is a device that allows the user or users to go forward or backwards through time and space.
A time machine is first used in "Mind Over Murder" when Stewie Griffin originally creates it to avoid teething but ends up going back when people copy his plans, which erases the events of the episode.
In "The Courtship of Stewie's Father", Stewie sets up a cutaway parodying Doc Brown at the end of the first film in the flying Delorean time machine telling Marty and Jennifer that they have to come to the future to fix their kids' situation.
In "Stu & Stewie's Excellent Adventure", the third segment of Stewie Griffin: The Untold Story, Stu Griffin, who is Stewie at 35 years old, uses a time machine in the form of a wristwatch to go on his vacations.
Peter drives the Delorean time machine in "The Perfect Castaway".
In "Road to Germany", Mort Goldman mistakes Stewie's new walk-in time machine for a bathroom stall. He lands in Poland on September 1, 1939, the beginning of World War II. When Stewie and Brian realize this, they go in to rescue him. This model is uranium powered.
In "The Big Bang Theory" Bertram used the time machine to try and kill Leonardo da Vinci, Stewie's ancestor in order to remove Stewie from the universe. However this in turn would have destroyed the universe so Stewie and Brian go back to prevent him from succeeding.
In the "Point of Stew" segment of "Family Guy Viewer Mail No. 2", Stewie uses the time machine to travel to April 5th, 1994 to stop Kurt Cobain from killing himself by convincing him to eat lots of Häagen-Dazs ice cream. In the present, Kurt is still alive but is morbidly obese.
In "Internal Affairs", Peter and Ernie the Giant Chicken fight their way onto Stewie's time pad and end up colliding with Marty McFly and the Delorean time machine in 1885 in Back to the Future Part III.
In "Yug Ylimaf", Brian uses Stewie's time machine without Stewie's permission to pick up women. To try to avoid getting caught, he attempts to reset the chronological gauge on the machine, but this ends up destroying the machine and reversing the flow of time.
Sick of close calls when they alter the past and are almost killed by Native Americans in "Life of Brian", Stewie destroys his time machine but is unable to rebuild it when he is unable to get parts after Brian is killed by a car. Later in "Christmas Guy", he spots himself buying a toy at the Quahog Mall and "borrows" the time machine return pad from his past self, going back in time to warn Brian and avoid his death.
Fed up with his family, Stewie builds an updated version of his time machine in "Chap Stewie" to change his conception to that of an upper-crust British family. But when he realizes he misses his old life, he sneaks into the University of Oxford and builds a rudimentary time machine that enables him to undo his original damage.
Stewie and Brian take Chris on a tour of history in the machine in "Stewie, Chris & Brian's Excellent Adventure" where Chris reveals that Brian had already told him about the machine while taking credit for it. When Chris becomes angry at being called a moron, he steals the return pad and boards the TITANIC, leaving Stewie and Brian to pursue him to recover it and get them off the ship. The pad becomes damaged, but Stewie is able to repair it once they manage to force their way onto a lifeboat.
Lois comes close to discovering Stewie's time machine in "Throw It Away", when she finds it and drags it out of his room, to declutter but Stewie's secret is kept safe when she assumes it's Meg's science project.
In "Baby Stewie", after failing to turn Stewie back to normal, Brian uses the time machine to go back and stop Stewie from building the machine that would alter his genes.
When Brian claims to be the smartest one in the family in "Who's Brian Now?", Stewie reminds him that he built a time machine.