Super Mario 64
Bugs/glitches

Crazy Bob-omb
If you didn't already know, you can pick up a Bob-omb, and put it down with the Z Button, and the Bob-omb will walk around casually until it explodes. In Course 1, climb the mountain and find the Bob-omb that is manning the water cannon (he is walking in place). Pick him up, then jump. While you're in mid-air, press Z, and this will leave the Bob-omb in mid-air! There is an interesting side-effect to this glitch... When he explodes, get the Coin that appears, then leave the area. When you come back, the Bob-omb will be there again, except he will never attack Mario! But, if you didn't get his Coin, then he will attack you.
-- [Sent in by LuigiHann]

Flying corpse  [screen shots: 1, 2]
If you manage to get Mario into a cannon hole when he has no power left, you'll still be able to shoot him out... but as soon as he lands, he will die. An easy place to perform this is in Course 2: Whomp's Fortress. Deplete your power meter to three pieces or less, jump in the cannon, and aim for the corner on the first wall sticking out. If you hit high enough off the corner, Mario will land back into the cannon with no energy.
An easier place is Course 8: Shifting Sand Land. Get down to at least 50% power, then jump in the cannon. Aim as far up and left as you can, then shoot out. Mario will hit an invisible wall and fall back in the cannon with no power. If you had a Wing Cap on, you can fly around the course with no power!
-- [Sent in by Dorian Blacquiere]

Out of the fire, into the water
In "Vanish Cap under the Moat," deplete your power meter to about two pieces. Let a flame set Mario on fire, and then run off the edge. If you time it right, Mario will have lost all power during his fall to the exit. As soon as he emerges into the waterfall outside the castle, he will drown.
-- [Sent in by Dorian Blacquiere]

Hat factory  [screen shot]
In Course 10, have the giant snowman blow off your hat. Don't pick it up, but instead go to the tree in the corner where the warp zone is. Warp back and forth a few times, then go back to your hat. For some reason, the more times you've warp, the more extra hats there will be. Now slowly approach the pile of hats to pick them up. Mario will pick up each hat and put it back on... except when he picks up the last one, he'll keep it in his hand! Mario can then run around holding his hat. Be careful, the game treats it like he doesn't have a hat, so he'll take more damage. To put the hat back on, simply warp, get a star, or die.

Hat factory: Wing Cap variation
In Course 8, let Klepto the Condor steal your hat. Climb the pillar and hit Klepto so the hat lands on the ground. Don't pick up the hat! Go to the tree by the oasis, warp back and forth a few times, then get the Wing Cap. Fly over to the stack of hats, and approach it the same way as the bug above. Mario will end up with the Wing Cap in his hand! And he can still fly; just do a triple jump to take-off.

Too quick  [screen shots: 1, 2]
In Course 1, pick Star 2 so Koopa the Quick appears. Get the Wing Cap, then do a triple jump starting from just past the first cannon hole. What you're trying to do here is land from your second jump and take-off really close to Koopa the Quick. If you took off from the right spot, and if you land from your flight far enough away (at least pass the first bridge), you'll get KtQ's race challenge message when you land. Accept it, and KtQ will never appear! If you mess up and get his message before you are able to take-off, decline his request and start over.
Using this bug, you can watch the timer max out at 9'59"99, with the race music playing forever. You can also finish the "race" in zero seconds flat! Here's how: Take-off as explained above, then when you're flying, swing around and land right in the first cannon hole (press Z for instant landing). Launch yourself to the sky island, and land in its cannon hole. Shoot yourself to the mountaintop, and land next to the flag. When you land, accept KtQ's race request. When the timer starts, it will immediately stop at 0'00"00 because you're already at the flag.

Through the Whomp
In Course 2, go up the tower until you run into a Whomp. Stand in front of him, and wait for him to start to fall on you. When he hops up, jump straight up before he falls. If you time it right, Mario will go right through the Whomp!

Forever Wing Cap music
In Course 1, pick Star 1 so the Big Bob-omb will be on top of the mountain. Get the Wing Cap, then quickly shoot yourself to the mountaintop via the cannons. Talk to the Big Bob-omb, then use the warp zone in the nearest cave. When you come out at the bottom, you won't have a Wing Cap but the flight music will keep playing... forever.
You can also do this in Star 2; get the Wing Cap, talk to Koopa the Quick, then warp. When the race is over, the flight music will start up again.

Instant death  [screen shot]
Get on the castle roof. Walk to the back right of the roof, through the red cone thing and all the way to the corner. (So if you're looking down at the field, you're on the south-west corner.) Try to get exactly on the corner, and then walk off the roof slowly so Mario catches the ledge and pulls himself back up. If you're back far enough, Mario will lose his hat and die when he gets back on the roof.

Through the bridge
Course 1, pick any Star. Follow the trail to the first bridge and go under it. Jump and hold A to grab on to the bottom. Continue to hold A as you use the Control Stick to navigate Mario to the lowest part of the bridge. Release A, and Mario will magically be pulled through to the top of the bridge.

Through the stairs  [screen shot]
Inside the Castle, go up the first staircase and through the door with a keyhole. You should be in a room with a spiral staircase. Go up to, but stay behind, the stone pillar thingie right before the stairs start. Hug the left wall, and double-jump. Mario will grab onto the ceiling and pull himself through the bottom of the staircase.

Boo!
Start one of your saved games. Inside the Castle, go up the first staircase and through the door with a keyhole. Turn around and go back to the room you were just in, then go in the room with Course 4. Walk down the ramp on the right, and you'll hear the screech of a Big Boo! (What really happened here is that you triggered the Big Boo in the hallway leading out to the courtyard. He won't be there now if you go in that hallway.)

Inside the Castle wall  [screen shots: 1, 2]
After you have 120 Stars, max out your power meter and go in the cannon outside the Castle. Shoot yourself onto the roof. Go around and get the Wing Cap, and fly back down to the cannon. Hop in the cannon and align the crosshairs as high as they go, but lined up with the Castle tower. Hold Down on the Control Stick, and press and hold A. After you shoot out of the cannon, keep holding Down and A. It may take a few tries, but you'll eventually land on the highest platform on the Castle tower. Make sure you're at the front part of the Castle, and walk straight at the wall. It may take a few tries, but eventually Mario will walk through the wall and drop down to the floor. You will now be in a weird room. It looks like the door will take you outside the Castle, but when you go in it, you end up inside the castle. Now the door looks like it will take you inside the Castle, but instead you end up back to where you started. To escape this loop: In the part where you can see the inside of the Castle, just walk through the wall and you'll be inside the Castle.

"Hat factory" and "Through the stairs" screen shots provided by David Dayton.