Stealing this list that I found while lurking Google.
These challenges are for any of the Pokemon games.
Nuzlocke Challenge -
The Nuzlocke Challenge is a challenge where you must release your Pokemon if they faint in battle. Also, you can only catch the first Pokemon you find on a Route. If all of your Pokemon must be released, you have to restart your game.
Monotype Challenge -
The Monotype Challenge is where you try to beat the whole game with a certain type of Pokemon and no other types are allowed!
Monocolor Challenge -
The Monocolor Challenge is where you only have a team of the same color. Shiny Pokemon do not count. Example: If you have a Shiny Scizor, it will still be treated as a red Pokemon instead of a green one.
Solo Run Challenge -
The Solo Run Challenge is where you beat the entire game with one Pokemon (HM Pokemon don't count). You can evolve this Pokemon if you want to.
All of these Challenges are good for any game!
I wanted to do some challenge runs through the various Pokemon gens (especially the Nuzlocke Challenge). Do you have any other suggestions for Pokemon challenges? List them here for future reference for me and any other runners.