  It is our intention to provide via fractional calculus a generalization of
the pure and compound Poisson processes, which are known to play a fundamental
role in renewal theory, without and with reward, respectively. We first recall
the basic renewal theory including its fundamental concepts like waiting time
between events, the survival probability, the counting function. If the waiting
time is exponentially distributed we have a Poisson process, which is
Markovian. However, other waiting time distributions are also relevant in
applications, in particular such ones with a fat tail caused by a power law
decay of its density. In this context we analyze a non-Markovian renewal
process with a waiting time distribution described by the Mittag-Leffler
function. This distribution, containing the exponential as particular case, is
shown to play a fundamental role in the infinite thinning procedure of a
generic renewal process governed by a power asymptotic waiting time. We then
consider the renewal theory with reward that implies a random walk subordinated
to a renewal process.
