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In order to have monsters in
Rune Factory, you have to capture them first. Hey, it's easier than buying farm animals, like in most
Harvest Moon games. To capture a monster you need the monster glove from
Tabatha. Once you have it just equip it in a cave like you would a weapon and walk up to the monster you want to befriend. They may fight you at first and cause some damage, but once you pet their heads they calm down and you are given the choice to either release them back to their world or send them to your monster huts.
You cannot capture bosses.