Mog is a recurring moogle name in Final Fantasy and they appear in Kingdom Hearts. It can refer to either male or female characters, and often moogles that play larger roles in the game.
In Final Fantasy VI, Mog is a named party member who can be recruited by the team depending on the player's choice. If the player choses to rescue Mog from a thief named Lonewolf, he will gladly join the party. This incarnation has the magical ability to summon the elements through magical dances that he learns. Recruiting him also allows players the option of later recruiting Umaro, a yeti who Mog has tamed and treats as an underling.
In Final Fantasy IX, Mog is Eiko's pet moogle.
In Final Fantasy XIII-2, a Moogle named Mog can turn into a bowsword for Serah Farron.