  This paper focuses on optimizing probabilities of events of interest defined
over general controlled discrete-time Markov processes. It is shown that the
optimization over a wide class of $\omega$-regular properties can be reduced to
the solution of one of two fundamental problems: reachability and repeated
reachability. We provide a comprehensive study of the former problem and an
initial characterisation of the (much more involved) latter problem. A case
study elucidates concepts and techniques.
