These are lecture notes on Grove and Petersen's work on manifolds with almost maximal volume with respect to a lower bound on curvature and an upper bound on radius. The notes correspond to three or four 50-minute lectures.
Manifolds with Almost Extremal Volume
Curtis Pro, Mike Sill, and I generalized the results and identified the diffeomorphism type of manifolds with almost maximal volume in
The diffeomorphism type of manifolds with almost maximal volume, with C. Pro & M. Sill, preprint.