  Given a compact Kahler manifold with an extremal metric (M,\omega), we give
sufficient conditions on finite sets points p_1,...,p_n and weights a_1,...a_n
for which the blow up of M at p_1,...,p_n has an extremal metric in the Kahler
class \pi^*[\omega] - \epsilon (a_1 PD[E_1] + .. + a_n PD[E_n]) for all
\epsilon sufficiently small. In particular our result implies that if
(M,\omega) is a toric manifold and p_1,...,p_n is any subset of the fixed locus
of the torus action, then such metrics exist for any choice of the weights. The
relationship with previous constructions of the first two authors for Kahler
constant scalar curvature metrics is discussed.
