  We prove the non-emptiness of $M_{H,Y}(v)$, the moduli space of
Gieseker-semistable sheaves on an unnodal Enriques surface $Y$ with Mukai
vector $v$ of positive rank with respect to a generic polarization $H$. This
completes the chain of progress initiated by H. Kim in \cite{Kim98}. We also
show that the stable locus $M^s_{H,Y}(v)\neq\varnothing$ for $v^2>0$. Finally,
we prove irreducibility of $M_{H,Y}(v)$ in case $v^2=0$ and $v$ primitive.

Algebraic Geometry