In this paper the first variational transition-state theory rate constant calculation for the OH + CH3SH reaction and several isotopic variants involving OD, CH3SD, and CD3SH is presented. Multidimensional tunneling corrections have been included when necessary. The potential energy surface has been described by low-level calculations at the MP2(full)/cc-pVDZ level combined with higher level calculations using the multilevel MCCM-CCSD(T)-CO-2m method. We have found that the reaction takes place by forming first a weakly bound complex that presents two hydrogen bonds between the hydroxyl radical and methanethiol. From this complex two different reaction pathways have been found: abstraction of the hydrogen atom attached to the sulfur atom and abstraction of the hydrogen atom of the methyl group. To take into account the different kinetic channels, the canonical and the competitive canonical unified statistical theories have been used to calculate the global rate constant of the perprotio reaction. Due to the slower nature of the H-abstraction of the methyl group, the global rate constant turns out to be equivalent to the overall rate constant for the abstraction of the hydrogen atom attached to the sulfur atom. It is only at the higher temperature range when a significant percentage of CH2SH is predicted. An activation energy of -0.54 kcal/mol in the range 225-430 K is obtained for the global reaction, in very good agreement with the experimental results. Several kinetic isotope effects of the global reactions have also been calculated and analyzed in view of previously published experimental results.