  We present a general framework to describe the evolutionary dynamics of an
arbitrary number of types in finite populations based on stochastic
differential equations (SDE). For large, but finite populations this allows to
include demographic noise without requiring explicit simulations. Instead, the
population size only rescales the amplitude of the noise. Moreover, this
framework admits the inclusion of mutations between different types, provided
that mutation rates, $\mu$, are not too small compared to the inverse
population size 1/N. This ensures that all types are almost always represented
in the population and that the occasional extinction of one type does not
result in an extended absence of that type. For $\mu N\ll1$ this limits the use
of SDE's, but in this case there are well established alternative
approximations based on time scale separation. We illustrate our approach by a
Rock-Scissors-Paper game with mutations, where we demonstrate excellent
agreement with simulation based results for sufficiently large populations. In
the absence of mutations the excellent agreement extends to small population
sizes.
