Mew glitch
The Mew Glitch is a software bug in the Generation I Game Boy Games. This glitch is famous because it allows you to get catch any Pokémon in the game, most notably Mew, without a GameShark or other hacking device -- and method #3 allows you to catch ANY Pokemon in the game (even rare, previously unknown glitchy Pokemon such as LM4 and X - x.)
Method #1
To acquire Mew at the earliest point possible in the game, one must not have defeated the Youngster on Route 25 or the Junior Trainer on Route 24 in the grass. It is prudent to save before doing this. Defeat Nugget Bridge as per the usual, then fight your way to the Youngster but do not fight him.
Return to Route 24 and catch (or trade over) an Abra. Place the Abra in your party, then return to Route 24 if you weren't there already. Stand above the Junior Trainer, but out of range of his sight. Move down until he sees you, but when he does, immediately press Start. Then use Teleport with your new Abra. If done correctly, the trainer should get the indicative exclamation mark as you Teleport.
After that, head north to the Youngster and have him spot you and then defeat him. Do not fight anything else, or talk to anyone before you do. (This includes the Youngster! You must engage in battle by him spotting you, and don't even walk up to him either. For some reason that causes a lock-up.)
After defeating the Youngster, teleport back to Cerulean again. Head north onto Nugget Bridge with some Balls, and when you get to the bridge, your start menu should pop up. Press B and you should engage in a fight against a level seven Mew, and it is quite easy to catch.
Method #2
Go to the gambler just east of Saffron City, in front of the Underground Tunnel entrance. Go north of him so that he is just off screen, and then as you walk south one step, just before he sees you, press start immediately. If he saw you, try over again. If it worked, the menu will appear and you will be able to Fly, Dig, or Teleport (flying is easiest). Fly to Cerulean City and while you are flying the exclamation point that usually appears ontop of a trainer will appear. Go to route 24 and defeat the trainer with one Slowpoke. Make sure he walks up to you, or the game will freeze. Defeat his Slowpoke, then fly to Lavender Town and walk west. Mew will appear.
Method #3
Go to the Gambler just east of Saffron City, OR go to either the biker in western route 14 or the one just below him (there are two of them together). Have the trainer off screen, walk up to him and immidiately press start before the trainer can see you. Then fly to Fucshia City. Go east to route 15 and make sure a Pokemon with a special stat of 21 is with you. Have that Pokemon first in your party. Battle a trainer (ANY trainer in the game) and then run into a wild Ditto, and make sure it transforms into your Pokemon with the 21 special stat. Then defeat the ditto. After this, do not run into any other wild Pokemon and do not battle any other trainers. Go back to the route where the trainer you originally evaded is located (for the gambler, east of Saffron, for the bikers, route 14). Press B and Mew will appear. Method #3 can be used to catch any Pokemon in the game, even Missingno. in the Yellow version. For example, in the Red/Blue versions, if a Pokemon with a special stat of 198 is used, a glitch called "LM4" will appear instead of Mew. (See [1])
(Important note: in the Red/Blue versions, Ditto will be found in Route 15. In the Yellow version, Ditto will be found in the Cinnabar Island Mansion's basement. Due to this, players in the Yellow version should NOT go to Route 15, but instead go to the Cinnabar Island Mansion's basement. It is recommened to bring an escape rope or repel to use immidiately after seeing Ditto, so that the player won't mess up the glitch by seeing the wrong Pokemon.)
Overview
The glitch in fact works as follows: Any trainer can be used for the first section (Junior Trainer), as long as the trainer can be engaged with by them getting an exclamation point above their heads. Any movement move (Teleport, Dig, Fly) can be used to trigger the glitch. For the Youngster, any Pokémon will do as long as it can be fought without walking up to a trainer. To finish off the glitch, you must return to the route of the trainer you "evaded". The Pokémon that you will find wild is dependent on the Special stat of the Pokémon you last faced (wild or from the trainer), and so using a Pokémon with the Special you need and finding a Ditto that transforms will produce an identical effect. For the record, the required Special stat for Mew is 21 (This is based on hexadecimal numbers, so you can get the appropriate number for a different Pokémon from a hacking device code list, then translate it to decimal).
