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.