This chapter is concerned with many-to-one functions of Markov processes. Neither the Markov property nor the Chapman-Kolmogorov equation are generally satisfied by the derived processes determined by such functions. The first section considers special circumstances under which the Chapman-Kolmogorov equation is still satisfied by the derived process. These involve conditions relating the structure of the function to the initially given Markov process. The case in which the Markov process is stationary and the transition function is self-adjoint or compact is discussed in some detail. The second section deals with preservation of the Markov property whatever the initial distribution. This involves the retention of a stronger property by the derived process. Finally, in the last section finite state non-Markovian processes are considered. The object is to determine which processes are instantaneous functions of finite state Markov chains. Algebraic concepts and tools are used to obtain a result of Heller.