This paper explores the coupled spin and valley physics in monolayers of MoS₂ and other group-VI dichalcogenides. The study shows that inversion symmetry breaking and spin-orbit coupling lead to coupled spin and valley physics, enabling control 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. Monolayer MoS₂ has two important distinctions from graphene. First, inversion symmetry is explicitly broken in monolayer MoS₂, which can give rise to the valley Hall effect where carriers in different valleys flow to opposite transverse edges when an in-plane electric field is applied. Second, MoS₂ has a strong spin-orbit coupling (SOC) originated from the d-orbitals of the heavy metal atoms, and can be an interesting platform to explore spin physics and spin-tronics applications absent in graphene due to its vanishing SOC. The study constructs a minimal band model on the basis of general symmetry consideration. The band structure of MoS₂, to a first approximation, consists of partially filled Mo d-bands lying between Mo-S s-p bonding and antibonding bands. The trigonal prismatic coordination of the Mo atom splits its d-orbitals into three groups. In the monolayer limit, the reflection symmetry in the z-direction permits hybridization only between A₁ and E orbitals, which opens a band gap at the K and -K points. The group of the wave vector at the band edges (K) is C₃h and the symmetry adapted basis functions are |ϕ_c⟩ = |d_z²⟩, |ϕ_v^τ⟩ = 1/√2(|d_x²−y²⟩ + iτ|d_xy⟩). The valence-band wave functions at the two valleys, |ϕ_v^+⟩ and |ϕ_v^−⟩, are related by time-reversal operation. The valley Hall and spin Hall effects are driven by the Berry phase associated with the Bloch electrons. The Berry curvature is defined by Ω_n(k) ≡ z · ∇_k × ⟨u_n(k)|i∇_k |u_n(k)⟩. For massive Dirac fermions described by the effective Hamiltonian, the Berry curvature in the conduction band is Ω_c(k)This paper explores the coupled spin and valley physics in monolayers of MoS₂ and other group-VI dichalcogenides. The study shows that inversion symmetry breaking and spin-orbit coupling lead to coupled spin and valley physics, enabling control 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. Monolayer MoS₂ has two important distinctions from graphene. First, inversion symmetry is explicitly broken in monolayer MoS₂, which can give rise to the valley Hall effect where carriers in different valleys flow to opposite transverse edges when an in-plane electric field is applied. Second, MoS₂ has a strong spin-orbit coupling (SOC) originated from the d-orbitals of the heavy metal atoms, and can be an interesting platform to explore spin physics and spin-tronics applications absent in graphene due to its vanishing SOC. The study constructs a minimal band model on the basis of general symmetry consideration. The band structure of MoS₂, to a first approximation, consists of partially filled Mo d-bands lying between Mo-S s-p bonding and antibonding bands. The trigonal prismatic coordination of the Mo atom splits its d-orbitals into three groups. In the monolayer limit, the reflection symmetry in the z-direction permits hybridization only between A₁ and E orbitals, which opens a band gap at the K and -K points. The group of the wave vector at the band edges (K) is C₃h and the symmetry adapted basis functions are |ϕ_c⟩ = |d_z²⟩, |ϕ_v^τ⟩ = 1/√2(|d_x²−y²⟩ + iτ|d_xy⟩). The valence-band wave functions at the two valleys, |ϕ_v^+⟩ and |ϕ_v^−⟩, are related by time-reversal operation. The valley Hall and spin Hall effects are driven by the Berry phase associated with the Bloch electrons. The Berry curvature is defined by Ω_n(k) ≡ z · ∇_k × ⟨u_n(k)|i∇_k |u_n(k)⟩. For massive Dirac fermions described by the effective Hamiltonian, the Berry curvature in the conduction band is Ω_c(k)