  Factorization is the central ingredient in any theoretical prediction for
collider experiments. We introduce a factorization formalism that can be
applied to any desired observable, like event shapes or jet observables, for
any number of jets and a wide range of jet algorithms in leptonic or hadronic
collisions. This is achieved by using soft-collinear effective theory to prove
the formal factorization of a generic fully-differential cross section in terms
of a hard coefficient, and generic jet and soft functions. In this formalism,
whether a given observable factorizes in the usual sense, depends on whether it
is inclusive enough, so the jet functions can be calculated perturbatively. The
factorization formula for any such observable immediately follows from our
general result, including the precise definition of the jet and soft functions
appropriate for the observable in question. As examples of our formalism, we
work out several results in two-jet production for both e+e- and pp collisions.
For the latter, we also comment on how our formalism allows one to treat
underlying events and beam remnants.
