  In this dissertation classification problems for K3-surfaces with finite
group actions are considered.
  Special emphasis is put on K3-surfaces with antisymplectic involutions and
compatible actions of symplectic transformations. Given a finite group $G$ it
is investigated if it can act in a compatible fashion on a K3-surfaces $X$ with
antisymplectic involution and to what extend the action of $G$ determines the
geometry of $X$. If $G$ has large order or rich structure a precise description
of $X$ as a branched double cover of a Del Pezzo surface is obtained. In
particular, this leads to classification results for the complete list of
eleven maximal finite groups of symplectic automorphisms of K3-surfaces.
  The remaining part of the thesis deals with generalizations to non-maximal
groups and K3-surfaces with non-symplectic automorphisms of higher order.
  The classification relies on an equivariant minimal model program for
surfaces, the study of finite group actions on Del Pezzo surfaces, and the
combinatorial intersection geometry of naturally associated exceptional and
fixed point curves.
