  We make precise the analogy between Goodwillie's calculus of functors in
homotopy theory and the differential calculus of smooth manifolds by
introducing a higher-categorical framework of which both theories are examples.
That framework is an extension to infinity-categories of the tangent categories
of Cockett and Cruttwell (introduced originally by Rosick\'y). The basic data
of a tangent infinity-category consist of an endofunctor, that plays the role
of the tangent bundle construction, together with various natural
transformations that mimic structure possessed by the ordinary tangent bundles
of smooth manifolds.
  The role of the tangent bundle functor in Goodwillie calculus is played by
Lurie's tangent bundle for infinity-categories, introduced to generalize the
cotangent complexes of Andr\'e, Quillen and Illusie. We show that Lurie's
construction admits the additional structure maps and satisfies the conditions
needed to form a tangent infinity-category which we refer to as the Goodwillie
tangent structure.
  Cockett and Cruttwell (and others) have started to develop various aspects of
differential geometry in the abstract context of tangent categories, and we
begin to apply those ideas to Goodwillie calculus. For example, we show that
the role of Euclidean spaces in the calculus of manifolds is played in
Goodwillie calculus by the stable infinity-categories. We also show that
Goodwillie's n-excisive functors are the direct analogues of n-jets of smooth
maps between manifolds; to state that connection precisely, we develop a notion
of tangent (infinity,2)-category and show that Goodwillie calculus is best
understood in that context.
