  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). A tangent
structure on an infinity-category X consists of an endofunctor on X, which
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 and that satisfy similar conditions.
  The tangent bundle functor in Goodwillie calculus is 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 on the
infinity-category of infinity-categories.
  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.
