  This is a survey on algorithmic questions about combinatorial and geometric
properties of convex polytopes. We give a list of 35 problems; for each the
current state of knowledege on its theoretical complexity status is reported.
The problems are grouped into the sections ``Coordinate Descriptions'',
``Combinatorial Structure'', ``Isomorphism'', ``Optimization'',
``Realizability'', and ``Beyond Polytopes''.
