  A $h$-instanton sheaf on a closed subscheme $X$ of some projective space
endowed with an ample and globally generated line bundle $\mathcal{O}_X(h)$ is
a coherent sheaf whose cohomology table has a certain prescribed shape. In this
paper we deal with $h$-instanton sheaves relating them to Ulrich sheaves.
Moreover, we study $h$-instanton sheaves on smooth curves and surfaces, cyclic
$n$-folds, Fano $3$-folds and scrolls over arbitrary smooth curves. We also
deal with a family of monads associated to $h$-instanton bundles on varieties
satisfying some mild extra technical conditions.
