  We prove the nonexistence of two-dimensional solitary gravity water waves
with subcritical wave speeds and an arbitrary distribution of vorticity. This
is a longstanding open problem, and even in the irrotational case there are
only partial results relying on sign conditions or smallness assumptions. As a
corollary, we obtain a relatively complete classification of solitary waves:
they must be supercritical, symmetric, and monotonically decreasing on either
side of a central crest. The proof introduces a new function which is related
to the so-called flow force and has several surprising properties. In addition
to solitary waves, our nonexistence result applies to "half-solitary" waves
(e.g. bores) which decay in only one direction.
