Application of the maintenance policies considered so far requires that the state of the system, operating or failed, is known at any time point. However, this may not be true when operating for instance fully automated systems, sensors or more complex monitoring systems. In such cases, inspections are indispensable maintenance measures. A standard situation, …rst formulated by R.E. Barlow and L.C. Hunter [15], is the following one: An undetected failure causes an economic loss which increases in time, whereas inspections are costly, too. What is the most cost-e¢ cient way to schedule inspections in time? This chapter deals with this situation and some of its recent rami…cations.