Almost linear actions by finite groups on S^{2n-1}

A free action of a finite group on an odd-dimensional sphere is said to be almost linear if the action restricted to each cyclic or 2-hyperelementary subgroup is conjugate to a free linear action. We begin this survey paper by reviewing the status of almost linear actions on the 3-sphere. We then discuss almost linear actions on higher-dimensional spheres, paying special attention to the groups SL_2(p), and relate such actions to surgery invariants. Finally, we discuss geometric structures on space forms or, more generally, on manifolds whose fundamental group has periodic cohomology. The geometric structures considered here are contact structures and Riemannian metrics with certain curvature properties.