Abstract: It is a general problem in Riemannian geometry to find and describe Riemannian manifolds that admit a metric of positive sectional curvature. All the known examples (which are relatively few in number) admit natural metrics with lots of symmetry. In this talk we will study certain classes of examples from the point of view of cohomogeneity of the metric; we compute the symmetry groups of the known examples of positively curved cohomogeneity one manifolds. As one application we construct examples of fundamental groups of positively curved manifolds that are not the fundamental groups of space forms. This is joint work with Karsten Grove and Wolfgang Ziller.