Which court handles maintenance?

Where a Family Court has been set up under the Family Courts Act, 1984, that's usually the forum for maintenance matters connected to marriage (like S.24/25 Hindu Marriage Act, 1955 claims). Where there's no Family Court, maintenance under Hindu Marriage Act/ Hindu Adoption and Maintenance Act goes to the District Court, while claims under Section 144 of Bhartiya Nagarik Suraksha Sanhita are handled by the Magistrate's court.