  We examine the surjectivity of isometries between weighted spaces of
holomorphic functions. We show that for certain classical weights on the open
unit disc all isometries of the weighted space of holomorphic functions, ${
\mathcal H}_{v_o}( \Delta)$, are surjective. Criteria for surjectivity of
isometries of ${ \mathcal H}_v(U)$ in terms of a separation condition on points
in the image of ${ \mathcal H}_{v_o}(U)$ are also given for $U$ a bounded open
set in $\mathbb{C}$. Considering the weight $v(z)= 1-|z|^2$ and the isomorphism
$f\mapsto f'$ we are able to show that all isometries of the little Bloch space
are surjective.
