  A covariant pseudodifferential calculus on Riemann surfaces, based on the
Krichever-Novikov global picture, is presented. It allows defining scalar and
matrix KP operators, together with their reductions, in higher genus. Globally
defined Miura maps are considered and the arising of polynomial or rational
${\cal W}$ algebras on R.S. associated to each reduction are pointed out. The
higher genus NLS hierarchy is analyzed in detail.
