Some new and recent results on the fixed point theory of multivalued contractions and nonexpansive mappings are presented. Discussions concerning Reich's problem are included. Existence of fixed points for weakly inward contractions is proved. Local contractions are also discussed. The Kirk-Massa theorem is extended to inward multivalued nonexpansive mappings. Using an inequality characteristic of uniform convexity, another proof of Lim's theorem on weakly inward multivalued nonexpansive mappings in a uniformly convex Banach space is included. The fixed point set function of a random contraction is proved to be measurable. Lim's fixed point theorem for nonexpansive self-mappings in a uniformly convex Banach space is randomized. Also, the fixed point set function of a single-valued random nonexpansive mapping in a uniformly smooth Banach space is shown to be measurable.