An inspection-maintenance policy is investigated for a system having various states. A policy is characterized by the type of maintenance and the next inspection time. Maintenance actions are classified into various types according to the depth of maintenance. Policy evaluation criterion is the expected cost accumulated up to the failure of the system. The problem is formulated as a Markov decision process and an optimal policy is found by using a policy improvement procedure. A numerical example illustrates the policy for a system having five states.