Alvastia Chronicles

The battle arena is located in Rubeus Town.  Initially, there are three cups: Bronze, Silver, and Gold.  Each level has three difficulty levels to clear.  You will always fight the same enemies in each difficulty level, but different parameters will be added to them depending on difficulty level (higher defense, poison damage every round, etc).  You must clear the difficulty levels in order in each cup, and each difficulty will show you a recommended party experience level - I suggest being several levels above the recommended level before attempting the battles.  You will also be required to pay an entrance fee each time you fight, so be aware of that (though if you've been following the walkthrough, you'll have more money than you know what to do with, so this should be no issue).

Once you've cleared all three cups, the Romance Cup will be available.  There's nothing different in the way it works as compared to the three battle cups before, but the highest level of it dictates that your vanguard (aka the front party, which if you haven't played around with character settings throughout the game, is Alan and his team) cannot act, so you'll definitely want to switch characters around before taking that level on, and as you'll only have eight fighters instead of twelve, you probably don't want to attempt this until you're well above the game's suggested level (which is 66).