Super Mario 64

For more information see the Wikipedia's article on Super Mario 64.

Inverted rooms
Once you have all 120 stars, a cannon near the pond in front of the castle will open and launch you to a roof, where Yoshi will be waiting. Also on the roof is a Wing Cap. There is a glitch that can be done with this:
 * Get the Wing Cap, then fly off the roof and into the cannon on the ground.
 * Launch towards the tower as high as it will go.
 * Keep flying towards it until you reach it. The upper part of the castle has 8 sides, so when you land on an outcrop face away from the wall that is wall that is 2nd to the left of the one that is in front (the "front" wall is the wall that was directly in front of you when you blasted from the cannon.)
 * Face away from the wall and then quickly turn around and run into it. You should fall down into a strange chamber. Remember that you'll lose half of your power.
 * You'll be in a chamber that is inverted-- in other words, the door that looks like it's going outside goes into the castle and vice versa. The only way out is to leave through a hidden wall in the chamber that's inside the castle.
 * When you fly to the castle outcrop, you can also walk straight up to the top of the castle instead of falling through (by walking up the 3rd left from the front wall)

Mutated Mario cap
(The teleporting part is normal, there are teleporters in other levels too. They are not considered glitches.)
 * Go to course 10 (Snowman's land)
 * Go to where the giant snowman is blowing and get blown off.
 * Your cap will fly off as always, but don't pick it up, go to the teleporter (it is a tree in the corner of the level, stand next to it for a few seconds and it will teleport you to another place.)
 * Teleport back to where the tree was.
 * Go to your cap. It will be mutated and glitchy. Pick it up and instead of going on Mario's head it will go in his hand.

Flying Mario corpse

 * Go to course 8 (Shifting Sand Land).
 * With the Pink Bomb-Omb, open up the cannon in the corner of the level.
 * Go in the cannon, and aim it just above the sun. (Move the cannon as far left and as far up as it will go.)
 * Fire the cannon, do it many times.
 * Eventually Mario's power will run out but he'll still be in the cannon.
 * If you fire straight out into the level Mario will technically be dead but still flying. If Mario lands though he'll die and if he gets coins they won't do anything (and if Mario lands in water he'll drown automatically.)