  We derive Bogomolny-type equations for the Abelian Higgs model defined on the
noncommutative torus and discuss its vortex like solutions. To this end, we
carefully analyze how periodic boundary conditions have to be handled in
noncommutative space and discussed how vortex solutions are constructed. We
also consider the extension to an $U(2)\times U(1)$ model, a simplified
prototype of the noncommutative standard model.
