Mewtwo is the strongest catchable Pokémon in the game and available to every player post-Elite Four. Mew requires a special event item — but there are ways to get both.
The Genetic Pokémon — the strongest non-event Pokémon in Gen I with 154 Sp. Attack. Available to all players after beating the Elite Four. Found in Cerulean Cave.
Cerulean Cave is locked until you beat the Elite Four AND obtain the National Pokédex via the Sevii Islands questline. The cave entrance northwest of Cerulean City will be blocked by a guard until both conditions are met.
Defeat the Elite Four and your rival in the Pokémon League. Then complete the Sevii Islands quest — deliver the Ruby to Celio on One Island and the Sapphire from the Rocket Warehouse on Five Island. Professor Oak will upgrade your Pokédex.
From Cerulean City, head west to Route 24, then northwest. The cave entrance is on the northwest shore of Cerulean. You'll need Surf to enter — the cave is partially flooded and requires swimming across underground lakes.
Cerulean Cave (also called Unknown Dungeon) is a multi-floor cave full of powerful wild Pokémon — Lickitung, Chansey, Ditto, and more. Navigate to the basement floor (B1F). The cave isn't overly complex but requires Surf to cross the water.
When you see Mewtwo's sprite at the end of the cave, save before approaching. Mewtwo is level 70 and has Recover — it will heal itself mid-battle, making the fight longer than expected.
Mewtwo is Psychic-type. It's weak to Bug, Ghost, and Dark moves. Use False Swipe to bring it to 1 HP (even with Recover, False Swipe can never knock it out). Apply sleep or paralysis, then throw Ultra Balls or your Master Ball.
Master Ball decision: Use it on Mewtwo if you've already caught your legendary dog without needing it. Save it for the dogs if you haven't hunted them yet — roaming Pokémon are harder to catch than stationary legendaries.
The New Species Pokémon — the original mystery. In FRLG, Mew is on Faraway Island, accessible only via the Old Sea Map event item. This event only ran in Japan and has ended.
Honest disclaimer: Mew requires the Old Sea Map, a Nintendo event item that was only distributed at Japanese events in 2005 and never officially released in English. There is no legitimate way to obtain Mew on an English cartridge without cheats. This is different from the "Mew under the truck" myth — that's not real.
The Old Sea Map was a Mystery Gift item that, when received, allowed the Vermilion Ferry to take you to Faraway Island — a hidden island in the Sevii Islands chain. Mew appears there in the tall grass at level 30.
If you're playing the Switch re-release, check the Mystery Gift menu before doing anything else. Nintendo has previously used re-releases as opportunities to distribute previously unavailable event Pokémon. There's a reasonable chance the Old Sea Map or Mew itself is available digitally.
With the Old Sea Map in Key Items, visit the Vermilion City harbor. The sailor takes you to Faraway Island — a lush island with tall grass. Mew will be hiding in the grass and running away (similar to the legendary dogs), but only within the island's grass patch — it can't escape the island.
For original GBA or emulator players without a legitimate Old Sea Map, this Action Replay code adds it to your inventory:
Enter this as a GameShark/Action Replay code. The Old Sea Map should appear in your Key Items. Then take the ferry from Vermilion to reach Faraway Island. Note: codes may vary by game version — verify for your specific setup before use.
The Mew Glitch (Gen 1 only): There's a famous glitch in original Red/Blue that allows catching Mew without cheats. This glitch does not work in Fire Red and Leaf Green — the games were specifically patched to remove it. Don't waste time trying it in FRLG.