1st Gen How to Catch Mew in Pokemon Red, Blue, and Yellow

[Mr. Sting]

How to Catch Mew in Pokemon Red/Blue/Yellow
Yes, believe it or not, it can be done without a GameShark or other cheating device. Here's what you'll need to do:

Restart your game and play until you get HM02 Fly from the house west of Celadon City.
DO NOT fight the following trainers:
The Youngster with a Slowpoke (Level 17) on the road to Bill's house (Route 25). He is the fourth trainer after you cross Nugget Bridge, standing directly above a female trainer and facing north.
The Gambler with two Poliwags and a Poliwhirl (all Level 22) between Lavender Town and Saffron City (Route 8). He is standing just east of the entrance to Saffron City, facing north toward the door of the Underground Path.
Carefully follow the series of steps illustrated below.

Make sure you have a Pokemon that knows Fly. On Route 8, stand in front of the door of the Underground Path. Save your game, since saving after this point will disrupt the procedure.
Walk down one step and immediately press Start. If you succeed, the menu will pop up before the Gambler "sees" you. Go to the Pokemon list and fly to Cerulean City.
The Gambler will see you just before you fly away.
Your Start button will no longer function. Go north across Nugget Bridge and head east until you find the trainer in this picture (he's the Youngster described above).
If you walk right up to the Youngster, the game will lock up, so you'll have to let him see you and walk up to you. Defeat his Slowpoke (afterwards, your Start button will work again), then fly to Lavender Town.
Head west into Route 8.
As soon as you enter the narrow path, the menu will pop up by itself. Press B to exit...
...and look who it is! Mew will be at Level 7 and its only attack will be Pound. Pokeball, go!

Once you've caught Mew, you can save your game without any ill-effects.

There are two alternative ways to enter Route 8 and trigger Mew after beating the Youngster. If you have access to Saffron City, you can fly there and go through the eastern city gate. Also, you can fly to Celadon City, head east to Route 7, enter the Underground Path, and emerge back in front of the Gambler.

 

[The Real AAA]

haha man i miss those late 90's early 00's nights lol

 

[Raskolnikov]

I remember well when I catch that Mew via glitch. Aaah gold days. :')

 

[Roaring Moon]

You can get Mew earlier than that (though it differs slightly between Red/Blue and Yellow). For this to work, you cannot have battled the Swimmer in Cerulean City's Gym, and you cannot have defeated Misty (because doing so makes the Swimmer unable to be battled):

Red/Blue

1. Play through the game until you clear the Nugget Bridge challenge.
2. Go to the grass to the left of the bridge and catch an Abra. Avoid the trainer in the grass! You don't want to battle him!
3. Heal at Cerulean City's Pokémon Center.
4. Head back to the aforementioned grass, but don't enter it!
5. Stand one tile above the right hand side of the grass and save, in case you screw up the next step.
6. Move one tile down and quickly open the start menu (before the trainer in the grass sees you).
7. Go into the party menu and use Abra's Teleport. You'll end up outside Cerulean City's Pokémon Center.
8. Go to the Cerulean City Gym and battle the Swimmer inside.
9. After defeating the Swimmer, head back to Nugget Bridge. If you did everything right, the start menu will open by itself. Close it to be thrust into a battle against a wild level 7 Mew!

Yellow

1. Play through the game until you defeat the Team Rocket Grunt in Cerulean City who stole the Dig TM.
2. Head down to Route 5 and catch an Abra.
3. Heal at Cerulean City's Pokémon Center.
4. Head to the grass to the left of Nugget Bridge, but don't enter it!
5. Stand one tile above the right hand side of the grass and save, in case you screw up the next step.
6. Move one tile down and quickly open the start menu (before the trainer in the grass sees you).
7. Go into the party menu and use Abra's Teleport. You'll end up outside Cerulean City's Pokémon Center.
8. Go to the Cerulean City Gym and battle the Swimmer inside.
9. After defeating the Swimmer, head back to Nugget Bridge. If you did everything right, the start menu will open by itself. Close it to be thrust into a battle against a wild level 7 Mew!

BONUS!

This can be done once you reach step 8, though it is purely optional.

1. Use Growl six times on the Swimmer's Shellder.
2. Defeat the Swimmer, head back to Nugget Bridge, and catch Mew. If step 1 was done right, it should be at level 1.
3. Place Mew in the front of the party.
4. Go to any patch of grass (those on Route 4 or earlier are recommended) with low-leveled wild Pokémon and get into a battle with any of them.
5. As soon as the battle begins, switch out Mew for another Pokémon in your party.
6. Defeat the wild Pokémon. If you did it right, Mew will instantly level up to level 100!

 

[Drogan]

It's amazing that this glitch stayed in the re-releases. Also it is possible to have 2 Mews before getting the Cascade Badge.