The use of probabilistic models for deterioration phenomena is presently gaining more and more ground. Typical practical applications are problems dealing with damage accumulation, as induced by e.g. chemical attack (carbonation, corrosion etc.), fatigue or time-dependent resistance (due to loss of prestressing etc.). In most cases the methods applied base on time-invariant reliability theory, treating time as a deterministic palrameter. There are, however, a number of cases, where such an approach would be simplistic and time-variant reliability theory has to be used instead: This is the case, as soon as random processes are involved. The present paper differentiates between the most important situations, starting from the most general case of non-smooth damage accumulation and proceeding to the description of several important special cases, for which it is possible to make some useful simplifications. Special attention is paid to the assumption of smooth deterioration, to models involving an initiation period and to the question, which reliability method has to be applied under which circumstances.