  We consider two-dimensional steady periodic gravity waves on water of finite
depth with a prescribed but arbitrary vorticity distribution. The water surface
is allowed to be overhanging and no assumptions regarding the absence of
stagnation points and critical layers are made. Using conformal mappings and a
new reformulation of Bernoulli's equation, we uncover an equivalent formulation
as "identity plus compact", which is amenable to Rabinowitz's global
bifurcation theorem. This allows us to construct a global connected set of
solutions, bifurcating from laminar flows with a flat surface. Moreover, a
nodal analysis is carried out for these solutions under a certain spectral
assumption involving the vorticity function. Lastly, downstream waves are
investigated in more detail.
