  We generalise Fl\o{}ystad's theorem on the existence of monads on the
projective space to a larger set of projective varieties. We consider a variety
$X$, a line bundle $L$ on $X$, and a base-point-free linear system of sections
of $L$ giving a morphism to the projective space whose image is either
arithmetically Cohen-Macaulay (ACM), or linearly normal and not contained in a
quadric. We give necessary and sufficient conditions on integers $a$, $b$, and
$c$ for a monad of type \[ 0\to(L^\vee)^a\to\mathcal{O}_{X}^{\,b}\to L^c\to0 \]
to exist. We show that under certain conditions there exists a monad whose
cohomology sheaf is simple. We furthermore characterise low-rank vector bundles
that are the cohomology sheaf of some monad as above.
  Finally, we obtain an irreducible family of monads over the projective space
and make a description on how the same method could be used on an ACM smooth
projective variety $X$. We establish the existence of a coarse moduli space of
low-rank vector bundles over an odd-dimensional $X$ and show that in one case
this moduli space is irreducible.
