It depends. Most duties are described in the terms of the lease. These duties may be assigned to either the landlord or tenant to repair, maintain, and pay utilities. In most states, landlords at a minimum have the legal duty to repair the premises so that the space is livable and the tenant has a duty to make ordinary repairs to maintain the property.
The information on this page is meant to provide a general overview of the law. The laws in your state and/or city may deviate significantly from those described here. If you have specific questions related to your situation you should speak with a local attorney.