  We show that inversion symmetry breaking together with spin-orbit coupling
leads to coupled spin and valley physics in monolayers of MoS2 and other
group-VI dichalcogenides, making possible controls of spin and valley in these
2D materials. The spin-valley coupling at the valence band edges suppresses
spin and valley relaxation, as flip of each index alone is forbidden by the
valley contrasting spin splitting. Valley Hall and spin Hall effects coexist in
both electron-doped and hole-doped systems. Optical interband transitions have
frequency-dependent polarization selection rules which allow selective
photoexcitation of carriers with various combination of valley and spin
indices. Photo-induced spin Hall and valley Hall effects can generate long
lived spin and valley accumulations on sample boundaries. The physics discussed
here provides a route towards the integration of valleytronics and spintronics
in multi-valley materials with strong spin-orbit coupling and inversion
symmetry breaking.
