I will outline Cheeger and Gromoll's 1972 proof of the Soul Theorem: Let M be a complete, noncompact manifold of nonnegative curvature. Then M has a soul, meaning a compact totally geodesic and totally convex submanifold S such that M is diffeomorphic to the normal bundle of S in M. Thus the study of manifolds with nonnegative curvature is reduced to the study of compact manifolds with nonnegative curvature. I will also present Perelman's beautiful and short 1994 proof that if M has a point where all sectional curvatures are positive then the soul of M consists of a single point.