We study functors from spaces to spaces or spectra that preserve
weak homotopy equivalences. For each such functor we construct a
universal n–excisive approximation, which may be thought of as its
n–excisive part. Homogeneous functors, meaning n–excisive
functors with trivial (n-1)–excisive part, can be classified: they
correspond to symmetric functors of n variables that are reduced and
1–excisive in each variable. We discuss some important examples,
including the identity functor and Waldhausen's algebraic K–theory.