  We derive an effective Hamiltonian which describes the dynamics of electrons
in the conduction band of transition metal dichalcogenides (TMDC) in the
presence of perpendicular electric and magnetic fields. We discuss in detail
both the intrinsic and the Bychkov-Rashba spin-orbit coupling (SOC) induced by
an external electric field.
  We point out interesting differences in the spin-split conduction band
between different TMDC compounds. An important consequence of the strong
intrinsic SOC is an effective out-of-plane $g$-factor for the electrons which
differs from the free-electron g-factor $g\simeq 2$. We identify a new term in
the Hamiltonian of the Bychkov-Rashba SOC which does not exist in III-V
semiconductors. Using first-principles calculations, we give estimates of the
various parameters appearing in the theory.
  Finally, we consider quantum dots (QDs) formed in TMDC materials and derive
an effective Hamiltonian which allows us to calculate the magnetic field
dependence of the bound states in the QDs. We find that all states are both
valley and spin split, which suggests that these QDs could be used as
valley-spin filters.
  We explore the possibility of using spin and valley states in TMDCs as
quantum bits, and conclude that, due to the relatively strong intrinsic
spin-orbit splitting in the conduction band, the most realistic option appears
to be a combined spin-valley (Kramers) qubit at low magnetic fields.
