In this paper we analyze, in detail, the magnetic properties of (Co/Ni) multilayers, a widely used system for spintronics devices. We use spin-polarized photoemission spectroscopy, magneto-optical Kerr effect, x-ray magnetic circular dichroism (XMCD), and anomalous surface diffraction experiments to investigate the electronic, magnetic, and structural properties in [Co(x)/Ni(y)] single-crystalline stacks grown by molecular-beam epitaxy. The spin polarization depends sensitively on the surface termination and for Co terminated stacks is found to be much larger than bulk Co, reaching at least 90% for 2 Co atomic planes. We observe a magnetization transition from in plane to out of plane when varying the Ni coverage on a Co layer in the submonolayer range, confirming the interface origin of the perpendicular magnetic anisotropy in this system. Angle-dependent XMCD using strong applied magnetic field allows us to show that the orbital magnetic moment anisotropy in Co is responsible for the anisotropy and that our results are consistent with Bruno's model. Surface x-ray diffraction shows that fcc stacking is preferred for 1-monolayer Co-based superlattices, whereas the hcp stacking dominates for larger Co thicknesses. We finally explored the role of the stacking sequence on the Co and Ni magnetic moments by ab initio calculations.