Monster can either be sent to your city by their corresponding god, or they can invade on their own. Usually, you can summon a hero to get rid of the monster.

Chimaera, the Sphinx, Echidna and the Harpies only appear in Poseidon.

MonsterGod who 'owns'
the monster
Hero who can kill it
Cyclops Zeus Odysseus
Kraken Poseidon Perseus
Cerberus Hades Hercules
Sphinx Hera Atalanta
Medusa Demeter Perseus
Hydra Athena Hercules
Calydonian Boar Artemis Theseus
Scylla Apollo Odysseus
Chimera Atlas Bellerophon
Dragon Ares Jason
Talos Hephaestus Jason
Hector Aphrodite Achilles
Minotaur Hermes Theseus
Maenads Dionysus Achilles
Echidna - Bellerophon
Harpies - Atalanta