A perfect game is when a pitcher allows no player to reach base, so they can't can't allow any hits, walks, or hit batsmen. So, by definition, a perfect game must be both a no-hitter and a shutout. 27 up, 27 down.
The feat has been achieved only 18 times in the history of major league baseball, 7 of which have happened in the 20 years I've been following the beautiful game.
On July 23, 2009, Chicago White Sox leftie pitcher Mark Buehrle pitched the 18th perfect game in major league history (his 2nd no-hitter). Amazingly it was the very first game he had worked with the catcher. He was assisted by a wall-climbing catch to record the first out in the 9th inning and rob a home run which would have ruined the perfect game.
On July 28, Buehrle followed up with another 5 2/3 innings of perfection to set the major league record for consecutive batters retired at 45.