  Let S be an orientable surface of finite type and let Mod(S) be its mapping
class group. We consider actions of Mod(S) by semisimple isometries on complete
CAT(0) spaces. If the genus of S is at least 3, then in any such action all
Dehn twists act as elliptic isometries. The action of Mod(S) on the completion
of Teichm\"uller space with the Weil-Petersson metric shows that there are
interesting actions of this type.
  Whenever the mapping class group of a closed orientable surface of genus g
acts by semisimple isometries on a complete CAT(0) space of dimension less than
g it must fix a point.
  The mapping class group of a closed surface of genus 2 acts properly by
semisimple isometries on a complete CAT(0) space of dimension 18.
