Electron capture and beta(-) decay are the dominant decay processes during late phases of evolution of heavy stars. Previous simulation results show that weak rates on isotopes of Molybdenum (Mo) have a meaningful contribution during the development of phases of stars before they go supernova. The relative abundance coupled with the stellar weak rates on Mo isotopes may change the lepton-to-baryon content of the core material. Here we report on the calculation of nuclear structure properties of Mo82-138 isotopes employing the RMF model. Later we calculate the weak decay rates of these isotopes. We use the pn-QRPA model to compute these rates. In the first step, the ground-state nuclear properties of Mo isotopes such as binding energy per nucleon, neutron and proton separation energies, charge radii, total electric quadrupole moments and deformation parameter of electric quadrupole moments have been calculated using density dependent version of RMF model with DD-PC1 and DD-ME2 functionals. The calculated electric quadrupole deformation parameters have been used in a deformed pn-QRPA calculation in the second phase of this work to calculate half-lives and weak decay rates for these Mo isotopes in stellar matter. We calculate the electron capture and beta-decay rates over an extensive range of temperature (0.01x10(9) K to 30x10(9) K) and density (10 to 10(11)) g/cm(3). Our study can prove useful for simulation of presupernova evolution processes of stars.