Complete Riemannian manifolds with nonnegative Ricci curvature have been well studied. Riemannian manifolds with a negative lower bound for Ricci curvature are considerably more complicated and less understood. I will first survey some recent results on such manifolds with positive bottom of spectrum. Then I will discuss a rigidity theorem which characterizes hyperbolic manifolds. The proof uses idea from potential theory and Brownian motion on Riemannian manifolds