  Two-dimensional (2D) transition-metal dichalcogenide (TMD) MX$_2$ (M = Mo, W;
X= S, Se, Te) possess unique properties and novel applications. In this work,
we perform first-principles calculations on the van der Waals (vdW) stacked
MX$_2$ heterostructures to investigate their electronic, optical and transport
properties systematically. We perform the so-called Anderson's rule to classify
the heterostructures by providing the scheme of the construction of energy band
diagrams for the heterostructure consisting of two semiconductor materials. For
most of the MX$_2$ heterostructures, the conduction band maximum (CBM) and
valence band minimum (VBM) reside in two separate semiconductors, forming type
II band structure, thus the electron-holes pairs are spatially separated. We
also find strong interlayer coupling at $\Gamma$ point after forming MX$_2$
heterostructures, even leading to the indirect band gap. While the band
structure near $K$ point remain as the independent monolayer. The carrier
mobilities of MX$_2$ heterostructures depend on three decisive factors, elastic
modulus, effective mass and deformation potential constant, which are discussed
and contrasted with those of monolayer MX$_2$, respectively.
