  We study the equilibrium domain structure and magnetic flux around a
ferromagnetic (FM) film with perpendicular magnetization M_0 on a
superconducting (SC) substrate. At 4{\pi}M_0<H_{c1} the SC is in the Meissner
state and the equilibrium domain width in the film, l, scales as
(l/4{\pi}{\lambda}_{L}) = (l_{N}/4{\pi}{\lambda}_{L})^{2/3} with the domain
width on a normal (non-superconducting) substrate, l_{N}/4\pi\lambda_L >> 1.
Here \lambda_L is the London penetration length. For 4{\pi}M_0 > H_{c1} and
l_{N} in excess of about 35 {\lambda}_{L}, the domains are connected by SC
vortices. We argue that pinning of vortices by magnetic domains in FM/SC
multilayers can provide high critical currents.
