  Atomically thin materials such as graphene and monolayer transition metal
dichalcogenides (TMDs) exhibit remarkable physical properties resulting from
their reduced dimensionality and crystal symmetry. The family of semiconducting
transition metal dichalcogenides is an especially promising platform for
fundamental studies of two-dimensional (2D) systems, with potential
applications in optoelectronics and valleytronics due to their direct band gap
in the monolayer limit and highly efficient light-matter coupling. A crystal
lattice with broken inversion symmetry combined with strong spin-orbit
interactions leads to a unique combination of the spin and valley degrees of
freedom. In addition, the 2D character of the monolayers and weak dielectric
screening from the environment yield a significant enhancement of the Coulomb
interaction. The resulting formation of bound electron-hole pairs, or excitons,
dominates the optical and spin properties of the material. Here we review
recent progress in our understanding of the excitonic properties in monolayer
TMDs and lay out future challenges. We focus on the consequences of the strong
direct and exchange Coulomb interaction, discuss exciton-light interaction and
effects of other carriers and excitons on electron-hole pairs in TMDs. Finally,
the impact on valley polarization is described and the tuning of the energies
and polarization observed in applied electric and magnetic fields is
summarized.
