A hypodifferential is a compact family of affine mappings that defines a local max-type approximation of a nonsmooth convex function. We present a general theory of hypodifferentials of nonsmooth convex functions defined on a Banach space. In particular, we provide complete characterizations of hypodifferentiability and hypodifferentials of nonsmooth convex functions, derive calculus rules for hypodifferentials, and study the Lipschitz continuity/Lipschitz approximation property of hypodifferentials that can be viewed as a natural extension of the Lipschitz continuity of the gradient to the general nonsmooth setting. As an application of our theoretical results we study the rate of convergence of several versions of the method of hypodifferential descent for nonsmooth convex optimization and present an accelerated version of this method having the faster rate of convergence \mathcal{O}(1/k^2) O ( 1 / k 2 ) .