  Monolayer transition metal dichalcogenides $MX_2$ ($M$ = Mo,W and $X$ = Te,
Se, S) in 1T' structure were predicted to be quantum spin Hall insulators based
on first-principles calculations, which were quickly confirmed by multiple
experimental groups. For a better understanding of their properties, in
particular their responses to external fields, we construct a realistic
four-band tight-binding (TB) model by combining the symmetry analysis and
first-principles calculations. Our TB model respects all the symmetries and can
accurately reproduce the band structure in a large energy window from -0.3 eV
to 0.8 eV. With the inclusion of spin-orbital coupling (SOC), our TB model can
characterize the nontrivial topology and the corresponding edge states. Our TB
model can also capture the anisotropic strain effects on the band structure and
the strain-induced metal-insulator transition. Moreover, we found that although
$MX_2$ share the same crystal structures and have the same crystal symmetries,
while the orbital composition of states around the Fermi level are
qualitatively different and their lower-energy properties cannot fully
described by a single k $\cdot$ p model. Thus, we construct two different types
of k $\cdot$ p model for $M$S$_2$,$M$Se$_2$ and $M$Te$_2$, respectively.
Benefiting from the high accuracy and simplicity, our TB and k $\cdot$ p models
can serve as a solid and concrete starting point for future studies of
transport, superconductivity, strong correlation effects and twistronics in
1T'-transition metal dichalcogenides.
