  Given a compact K\"ahler manifold $X$, a quasiplurisubharmonic function is
called a Green function with pole at $p\in X$ if its Monge-Amp\`ere measure is
supported at $p$. We study in this paper the existence and properties of such
functions, in connection to their singularity at $p$. A full characterization
is obtained in concrete cases, such as (multi)projective spaces.
