  In this paper we consider the Skorokhod embedding problem for target
distributions with non-zero mean. In the zero-mean case, uniform integrability
provides a natural restriction on the class of embeddings, but this is no
longer suitable when the target distribution is not centred. Instead we
restrict our class of stopping times to those which are minimal, and we find
conditions on the stopping times which are equivalent to minimality. We then
apply these results, firstly to the problem of embedding non-centred target
distributions in Brownian motion, and secondly to embedding general target laws
in a diffusion. We construct an embedding (which reduces to the Azema-Yor
embedding in the zero-target mean case) which maximises the law of \sup_{s \le
T} B_s among the class of minimal embeddings of a general target distribution
\mu in Brownian motion. We then construct a minimal embedding of \mu in a
diffusion X which maximises the law of \sup_{s \le T} h(X_s) for a general
function h.
