  Semi-Markov processes are a generalization of Markov processes since the
exponential distribution of time intervals is replaced with an arbitrary
distribution. This paper provides an integro-differential form of the
Kolmogorov's backward equations for a large class of homogeneous semi-Markov
processes, having the form of an abstract Volterra integro-differential
equation. An equivalent evolutionary (differential) form of the equations is
also provided. Fractional equations in the time variable are a particular case
of our analysis. Weak limits of semi-Markov processes are also considered and
their corresponding integro-differential Kolmogorov's equations are identified.
