  We perform a systematic investigation of Kazhdan's relative Property (T) for
pairs (G,X), where G a locally compact group and X is any subset. When G is a
connected Lie group or a p-adic algebraic group, we provide an explicit
characterization of subsets X of G such that (G,X) has relative Property (T).
In order to extend this characterization to lattices of G, a notion of
"resolutions" is introduced, and various characterizations of it are given.
Special attention is paid to subgroups of SU(2,1) and SO(4,1).
