  We develop a global theory for complete hypersurfaces in $\mathbb{R}^{n+1}$
whose mean curvature is given as a prescribed function of its Gauss map. This
theory extends the usual one of constant mean curvature hypersurfaces in
$\mathbb{R}^{n+1}$, and also that of self-translating solitons of the mean
curvature flow. For the particular case $n=2$, we will obtain results regarding
a priori height and curvature estimates, non-existence of complete stable
surfaces, and classification of properly embedded surfaces with at most one
end.
