  A comprehensive approach to Sobolev-type embeddings, involving arbitrary
rearrangement- invariant norms on the entire Euclidean space R^n, is offered.
In particular, the optimal target space in any such embedding is exhibited.
Crucial in our analysis is a new reduction principle for the relevant
embeddings, showing their equivalence to a couple of considerably simpler
one-dimensional inequalities. Applications to the classes of the Orlicz-Sobolev
and the Lorentz-Sobolev spaces are also presented. These contributions fill in
a gap in the existing literature, where sharp results in such a general setting
are only available for domains of finite measure.
