  In the development of stochastic integration and the theory of
semimartingales, Markov processes have been a constant source of inspiration.
Despite this historical interweaving, it turned out that semimartingales should
be considered the `natural' class of processes for many concepts first
developed in the Markovian framework. As an example, stochastic differential
equations have been invented as a tool to study Markov processes but nowadays
are treated separately in the literature. Moreover, the killing of processes
has been known for decades before it made its way to the theory of
semimartingales most recently. We describe, when these and other important
concepts have been invented in the theory of Markov processes and how they were
transferred to semimartingales. Further topics include the symbol,
characteristics and generalizations of Blumenthal-Getoor indices. Some
additional comments on relations between Markov processes and semimartingales
round out the paper.

Probability