  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.
