  I discuss the so-called stochastic individual based model of adaptive
dynamics and in particular how different scaling limits can be obtained by
taking limits of large populations, small mutation rate, and small effect of
single mutations together with appropriate time rescaling. In particular, one
derives the trait substitution sequence, polymorphic evolution sequence, and
the canonical equation of adaptive dynamics. In addition, I show how the escape
from an evolutionary stable conditions can occur as a metastable transition.
This is a review paper that will appear in
  "Probabilistic Structures in Evolution", ed. by E. Baake and A. Wakolbinger.
