We study Gamow-Teller strength distributions within a QRPA approach performed on top of a deformed Skyrme-Hartree-Fock+BCS single particle basis with the inclusion of residual spin-isospin interactions in both particle-hole and particle-particle channels. We focus our attention in several problems of interest in Nuclear Structure and Nuclear Astrophysics, such as the β-decay properties of proton-rich medium-mass nuclei of astrophysical interest and the deformation dependence of the Gamow-Teller strength distributions in neutron deficient Pb isotopes. We also discuss the role of deformation in the two-neutrino double beta decay.