  We study abelian BPS vortices on a surface $S$ with boundary, which satisfy
the Neumann boundary condition on the norm of the scalar field, or
equivalently, that the current along the boundary vanishes. These vortices have
quantised magnetic flux and quantised energy. Existence of such vortices is
manifest when $S$ is the quotient by a reflection of a smooth surface without
boundary, for example a hemisphere. The $N$-vortex moduli space then admits an
interesting stratification, depending on the number of vortices in the interior
of $S$ and the number of half-vortices on the boundary.
