Let G be a locally compact group acting by measure-preserving transformations on a probability space (X,mu). To every probability measure on G there is an associated averaging operator on L^p(X,mu). Ergodic theorems describe the pointwise and norm limits of sequences of such operators. In joint work with Amos Nevo, we develop a new general approach based on reducing the problem to the amenable case. From this we obtain ergodic theorems for sector and spherical averages when G is a rank 1 Lie group or a countable Gromov hyperbolic group.