Mew is a mythical Pokémon which you can now catch in Pokémon Go. It isn't available in the wild, though. You must use the new Research feature in the game, accessed by tapping the binoculars icon on the main screen.
On the right-hand tab is Special Research. This is given by Professor Willow who, thus far, hasn't really been present in Pokémon Go aside from his very basic guide for beginners.
Special Research is much more involved than Field Research, and is the only way to catch Mew. Note that the progress bar at the top of the screen is nothing to do with Special Research - you get a stamp for completing Field research.
To catch Mew, you must progress through eight stages of 'Special Research'. The initial tasks include transferring five Pokémon to the Professor, spinning five PokéStops and catching 10 Pokémon.
These quests all reward you with 500XP, but are grouped and must be completed before you can progress to stage 2. When you've claimed all three rewards, scroll down and collect the bonus for completing that part, which includes a three-use incubator:
Once you've made it through all eight stages, you get a chance to catch Mew. Don't worry: the catch rate for Mew is 100% so it won't run away.
The tricky part is that Mew appears for only a second and then becomes 'invisible'. You can just about make out its shadow, but it's much easier if you stand in the shade as phone screens aren't really bright enough in sunlight.
A Mythical Discovery: tasks and rewards
Here are the tasks you need to complete in each stage on A Mythical Discovery:
- Spin 5 PokéStops
- Catch 10 Pokémon
- Transfer 5 Pokémon
Rewards: 10x Great Balls, 1x incubator, 3x lure modules
- Earn 2 candies walking your buddy
- Make 10 Great throws
- Hatch 3 eggs
Rewards: 2000 Stardust, 3x incense, 20x Great balls
- Reach level 15
- Battle in a gym 2 times
- Battle in 2 raids
Rewards: 1x Fast Technical Machine, 1x Charged Technical Machine, 2x Star Pieces
- Earn a Silver Kanto medal
- Evolve 20 Pokémon
- Earn 5 candies walking your buddy
Rewards: 4000 Stardust, 3x lure modules, 20x Great balls
- Catch Ditto
- Make 20 Great throws
- Catch 10 Ghost-type Pokémon
Rewards: 1x Premium raid pass, 1x lucky egg, 15x revives
- Reach level 25
- Battle 10 raids
- Evolve a Magikarp (400 candies required)
Rewards: 6000 Stardust, 5x rare candy, 3x incense
- Catch 50 Pokémon using a berry
- Make 1 Excellent curve throw
- Earn a Gold Kanto medal
Rewards: 8000 Stardust, 5x Ultra balls
- Catch Mew
Rewards: 10000 Stardust, 1x Super Incubator, 20x Mew candy