If you're playing "Dragon Quest VIII: Journey of the Cursed King," you will stumble across the Monster Arena early in the game. It is located on the southeast continent, southeast of the town of Pickham. You can play a mini-game at the Monster Arena once it has been unlocked. To unlock it, speak with Morrie at the arena and he will ask you to scout three "infamous monsters" for him. Additionally, there are over 80 other infamous monsters that can be scouted; these are not required to unlock the arena. They are distributed on all eight island continents on the world map.
Unlocking Monster Arena: Slimy Smiley (Smiles)
Slimy Smiley is just to the south on the path at the entrance of Trodain Castle. Trodain Castle is on the bottom half of the northeast continent, which is divided by a river. After crossing into the southern half of the continent, head to the northwest corner; Trodain Castle will be by the river.
Unlocking Monster Arena: Hollow Knight (Lonely Joe)
Hollow Knight can be found between the Ruined Abbey and Maella Abbey. Both abbeys are located in the southern half on the southeast continent, near the river. Hollow Knight will be to the east and north of the path south from Maella Abbey to Simpleton. He'll be right next to the river.
Unlocking Monster Arena: Squiggly Squiggler (Squiggles)
Squiggly Squiggler can be found south of Red's Den, on the beach. Red's Den is south of Pickham on the southeast continent. Head south from the Monster Arena until you hit the coast and follow the coast southwest. You'll eventually run into Squiggly Squiggler.
Other Monsters: Octurion
Octurion is located on the southwest continent. Head south and follow the river from Argonia, the town in the middle of the continent. Octurion can be found on the beach south of where the river flows into the ocean.
Other Monsters: Sugi
Sugi can be found on the largest island in the southern part of the world map. Head southwest from Empycchu towards the mountain range, and Sugi will be slightly to the west and slightly north of the mountain range. Sugi can only be caught at night.
Other Monsters: Steropes
There is a small island to the west in the channel between the northeast and southeast continents. On this island is Princess Minnie's Castle. Directly west of the castle is Steropes, who will be hard to miss because the island is very small. He will be between the castle and the coast.