  Molybdenum disulfide, MoS2, has recently gained considerable attention as a
layered material where neighboring layers are only weakly interacting and can
easily slide against each other. Therefore, mechanical exfoliation allows the
fabrication of single and multi-layers and opens the possibility to generate
atomically thin crystals with outstanding properties. In contrast to graphene,
it has an optical gap of 1.9 eV. This makes it a prominent candidate for
transistor and opto-electronic applications. Single-layer MoS$_2$ exhibits
remarkably different physical properties compared to bulk MoS$_2$ due to the
absence of interlayer hybridization. For instance, while the band gap of bulk
and multi-layer MoS$_2$ is indirect, it becomes direct with decreasing number
of layers. In this review, we analyze from a theoretical point of view the
electronic, optical, and vibrational properties of single-layer, few-layer and
bulk MoS$_2$. In particular, we focus on the effects of spin-orbit interaction,
number of layers, and applied tensile strain on the vibrational and optical
properties. We examine the results obtained by different methodologies, mainly
ab initio approaches. We also discuss which approximations are suitable for
MoS$_2$ and layered materials. The effect of external strain on the band gap of
single-layer MoS$_2$ and the crossover from indirect to direct band gap is
investigated. We analyze the excitonic effects on the absorption spectra. The
main features, such as the double peak at the absorption threshold and the
high-energy exciton are presented. Furthermore, we report on the phonon
dispersion relations of single-layer, few-layer and bulk MoS$_2$. Based on the
latter, we explain the behavior of the Raman-active $A_{1g}$ and $E^1_{2g}$
modes as a function of the number of layers.
