In this book invariant probabilities for a large class of discrete-time homogeneous Markov processes known as Feller processes are discussed. These Feller processes appear in the study of iterated function systems with probabilities, convolution operators, certain time series, etc. Rather than dealing with the processes, the transition probabilities and the operators associated with these processes are studied.Main features:- an ergodic decomposition which is a "reference system" for dealing with ergodic measures- "formulas" for the supports of invariant probability measures, some of which can be used to obtain algorithms for the graphical display of these supports- helps to gain a better understanding of the structure of Markov-Feller operators, and, implicitly, of the discrete-time homogeneous Feller processes- special efforts to attract newcomers to the theory of Markov processes in general, and to the topics covered in particular- most of the results are new and deal with topics of intense research interest.