There are currently two challenges in the game.
Pupil pupil's challenge. Basically you reset all your modifiers to 0% and climb from the bottom once more. when you complete it unlocks an autofight for challenger battles each challenge complete gives 10% of the rewards of a regular battle. maxing at 10 challenges for 100% rewards.
Challengers challenge. Beat a challenger with every theme (13 themes) the level required is equal to challengers challenge completed + 1. Each challenge you beat gives you a 10% chance to earn a bonus medal from challengers. capping at 20 challenges for a guaranteed 2 bonus medals. the order is the same as the order on the theme research tab