This paperconcerns a Markov operator $T$ on a space $L_1$, and a Markov process $P$ which defines a Markov operator on aspace $M$ of finite signed measures. For $T$, the paper presentsnecessary and sufficient conditionsfor: \begin{enumerate}\item [(a)] the existence of invariant probability densities (IPDs)\item [(b)] theexistence ofstrictly positive IPDs, and\item [(c)] the existence and uniqueness ofIPDs.\end{enumerate} Similar results on invariant probability measures for $P$ are presented. The basicapproach is to pose a fixed-point problem as the problem of solving a certain linear equation in a suitableBanach space, and then obtain necessary and sufficient conditions for this equation to have a solution.

1991 Mathematics Subject Classification: 60J05, 47B65, 47N30.