  Two-dimensional (2D) transition metal dichalcogenide (TMD) nanosheets exhibit
remarkable electronic and optical properties. The 2D features, sizable
bandgaps, and recent advances in the synthesis, characterization, and device
fabrication of the representative MoS$_2$, WS$_2$, WSe$_2$, and MoSe$_2$ TMDs
make TMDs very attractive in nanoelectronics and optoelectronics. Similar to
graphite and graphene, the atoms within each layer in 2D TMDs are joined
together by covalent bonds, while van der Waals interactions keep the layers
together. This makes the physical and chemical properties of 2D TMDs layer
dependent. In this review, we discuss the basic lattice vibrations of
monolayer, multilayer, and bulk TMDs, including high-frequency optical phonons,
interlayer shear and layer breathing phonons, the Raman selection rule,
layer-number evolution of phonons, multiple phonon replica, and phonons at the
edge of the Brillouin zone. The extensive capabilities of Raman spectroscopy in
investigating the properties of TMDs are discussed, such as interlayer
coupling, spin--orbit splitting, and external perturbations. The interlayer
vibrational modes are used in rapid and substrate-free characterization of the
layer number of multilayer TMDs and in probing interface coupling in TMD
heterostructures. The success of Raman spectroscopy in investigating TMD
nanosheets paves the way for experiments on other 2D crystals and related van
der Waals heterostructures.
