This paper concerns a Markov operator T on a space L I , and a Markov process P, which defines a Markov operator on a space M of finite signed measures. For T, the paper presents necessary and suf ic ient conditions for: (a) the existence of invariant probability densities (IPDs) (b) existence of strictly positive IPDs, and (c) existence and uniqueness of IPDs. Similar results on invariant probability measures for P are presented. The basic approach is to pose a fixed-point problem as the problem of solving a certain linear equation in a suitable Banach space, and then obtain necessary and sufficient conditions for this equation to have a solution.