The spectral geometry of negatively curved manifolds has received more attention than its positive curvature counterpart. In this paper we will survey a variety of spectral geometry results that are known to hold in the context of hyperbolic manifolds and discuss the extent to which analogous results hold in the setting of spherical manifolds. We conclude with a number of open problems.

The authors wishes to express their thanks to the referee for several helpful comments. Furthermore, they are indebted to Loren Spice by a very useful answer in Mathoverflow to a question made by the first named author.