The Mew glitch is a software bug in all three of the Generation I Game Boy games, discovered in 2003. This glitch is famous for allowing players to catch any non-glitched Pokémon in the game, most notably Mew. Method #3 of the Mew glitch allows one to catch both glitch Pokémon and regular Pokémon.
This is the Mew glitch.
To get Mew at the earliest point possible in the game, the player must not have defeated the Swimmer in Cerulean Gym and the Jr. Trainer (the one who says "I saw your feat from the grass" when the battle with him begins) on Route 24 in the grass. The player must defeat the Trainers on Nugget Bridge as usual.
The player must have an Abra that can Teleport in their party, and therefore, if playing Pokémon Yellow, must have reached Route 5 to catch one or traded for one beforehand.
The player must first heal at the Cerulean City Pokémon Center, so they may Teleport there later. With an Abra in their party, the player then needs to stand north of the Jr. Trainer so that he is just below the screen. It is recommended that the player now saves the game.
The player must move down until seen and immediately press the "START" button. The player then needs to Teleport with Abra. If done correctly, the Jr. Trainer should get the indicative exclamation mark above his head immediately before the player Teleports. At this point, the menu will not work until after the player enters a battle, though it is imperative to not battle.
The player then needs to head east from the Pokémon Center into the Cerulean Gym and battle the Swimmer. After defeating the Swimmer (the menu should be available upon victory), the player then must head to Route 24, and after several steps the menu will appear. Upon closing of the menu, a battle will begin with a wild Mew.
This is Method 1 of obtaining it via this glitch.
The player must follow the steps for Method #1 exactly until after teleporting from the Jr. Trainer. The player then must head north to Route 25 and battle the Youngster. As in Method #1, it is imperative to avoid all other battles before then. The player must engage in battle by being seen (and must walk along the wall to be seen, not towards him, otherwise the game will freeze).
After defeating the Youngster, the player then needs to walk towards Cerulean City. After a certain amount of steps the Start menu will appear. Closing it should begin a battle against a wild Mew.
This is an alternate Method 1.
For this method, the player needs to find any Trainer in the game that will engage in battle the moment the Trainer is on screen and the player is in the Trainer's line of sight. The player then needs to take one step down closer to the NPC, but make sure that the NPC does not see the player, and then immediately press the "START" button. The menu will appear and the player will be able to Fly or Teleport. The player needs to fly or teleport to Cerulean City. While flying, the exclamation point that usually appears above a Trainer's head when they challenge the player to battle will appear. The player then needs to go to Route 24 and, defeat the Youngster that owns a Slowpoke (and must walk along the wall to be seen, not towards him or the game will freeze). After defeating him, the player needs to fly or teleport to Lavender Town and head to Route 8. The menu will appear. Pressing B will make a wild Mew appear at Level 7.
For this method, the player first needs a Pokémon with a Special stat of 21. It doesn't matter if the stat was naturally leveled or boosted with stat experience, as long as it's not due to an in-battle Special-stat boosting skill. The player needs to find any Trainer in the game that will engage in battle the moment the Trainer is on screen and the player is in the Trainer's line of sight. The player then needs to take one step down closer to the Trainer, but make sure that the Trainer does not see the player, and then immediately press the "START" button. Once the player has pressed Start and the menu appears, the player needs to Fly or Teleport elsewhere.
The location being traveled to must have a path via which the player can walk both to a place where there are wild Ditto (Route 15RB or Pokémon MansionY) and a Trainer to fight. This is because the Start button will no longer work until after a battle with a Trainer (to be mentioned) because the game has been tricked into thinking it's in battle mode. Once the battle with the Trainer the player picked (the first Trainer) was evaded successfully with Fly or Teleport, the player then needs to battle any Trainer in the new location; there needs to be one space between the player and the second Trainer when the battle is initiated so that the second Trainer has to walk up to the player. After defeating the Trainer, the player needs to encounter a wild Ditto, and battle it until it uses Transform on the player's Pokémon with the Special Stat of 21.
After defeating the Ditto (with any of the player's Pokémon), the player must not battle, and go back to the location where the first Trainer the player first evaded is located. Once the location is approached, the menu will automatically appear. The player must press the B button. A battle will begin with a wild Mew.
This method can be used to catch any Pokémon in the game, even some glitch Pokémon. While a Pokémon with a special stat of 21 will give one Mew, using a Pokémon with a different special stat will give the player a different Pokémon. For example, in the Red/Blue versions, if a Pokémon with a Special stat of 198 is used, ゥL ゥM 4 will appear instead of Mew. Some Special stats will cause glitch Trainers to appear.
Method 3 also known as the Ditto Glitch
Hope I helped!