There are 4 Colosseums in Blunder Hills, Yum-Yum Desert, Frostbite Tundra and Hyperion Nebula. Each Colosseum costs different amounts of Colosseum Ticket to enter. In the Colosseum, you fight waves of enemies for rewards.

All Colosseums feature the same basic mechanics:

You have 5 minutes to complete the Colosseum. Whether you clear all waves or run out of time, you'll receive reward chests at the end based on your final score.
There are 15 waves of monsters. The final wave includes a boss. All 15 waves + the boss must be cleared in order to earn Golden Chest rewards.
Pressing 'Next Round' will progress to the next round immediately, and increase your score multiplier. However, the more enemies there are on the screen, the faster your multiplier will decrease! You must defeat them quickly to maintain your score multiplier.
Getting multi-kill combos also contributes to your score multiplier.
Clearing the full Colosseum with time remaining will earn you a Time Bonus, increasing your score.

Any increasing Mob's Health curse, such as Big Brain Time, causes slowing down wave and takes longer to complete Colosseum.
Use your active attacks (the ones you equip to your attack bar) strategically to take out waves. Turning off auto or turning off the "Use attacks in auto" option will ensure the AI doesn't waste your strongest moves.
If your damage and strategy are solid, increasing your Move Speed can help you finish within the time limit/set new high scores. If you don't need healing food/damage food, swap in Speed Potions.
If you die in the Colosseum and don't have an Insta-Rez remaining, your Town Respawn will let you collect your rewards.
It is best to use your Divine Knight for Colosseum die to its good AOE and drop rate multipliers from the Orb Of Remembrance Talent.