How to Catch Uxie, Mesprit, and Azelf in Pokémon Black 2 and White 2

Steps

Beat the Elite 4 and Champion. You need to do this before you're able to catch these Pokémon.

Go west of Virbank City to Route 20.

Go to the southern part of the route (near the dark grass) and use Surf. Go south and then west until you find a waterfall. Use Waterfall and go west until you reach a fork. Go south, then west until you find a cave. Enter it.

You'll trigger a cutscene where Uxie, Mesprit, and Azelf appear, and they'll head off to three different locations.

Finding Uxie

Go to Nacrene City.

Go to the museum to the north. There, go into the middle of the circular area, and a text box will appear asking if you want to check the area. Say yes, and Uxie will appear.

Talk to it to battle it.

Finding Mesprit

Go to Celestial Tower, which is located on Route 7, north of Mistralton City.

Go up the tower until you reach the top, where the bell is.

Go to the area just before the stairs, and a text box will appear asking if you want to check the area. Say yes, and Mesprit will appear.

Talk to it to battle it.

Finding Azelf

Go to Route 23, west of the Giant Chasm. You'll want to go there by flying to Victory Road, and then going south to Route 23.

Go down the stairs, then go west. Cut down the bush using Cut, then go west until you reach a plateau with a large boulder.

Go up the stairs, and a text box will appear asking if you want to check the area. Say yes, and Azelf will appear.

Talk to it to battle it.