In this dissertation, we provide an accurate description of spin transport in magnetic textures and in particular, we investigate in detail, the nature of spin torque and magnetic damping in such systems. Indeed, as will be further discussed in this thesis, the current-driven velocity of magnetic textures is related to the ratio between the so-called non-adiabatic torque and magnetic damping. Uncovering the physics underlying these phenomena can lead to the optimal design of magnetic systems with improved efficiency. We identified three interesting classes of systems which have attracted enormous research interest (i) Magnetic textures in systems with broken inversion symmetry: We investigate the nature of magnetic damping in non-centrosymmetric ferromagnets. Based on phenomenological and microscopic derivations, we show that the magnetic damping becomes chiral, i.e. depends on the chirality of the magnetic texture. (ii) Ferromagnetic domain walls, skyrmions and vortices: We address the physics of spin transport in sharp disordered magnetic domain walls and vortex cores. We demonstrate that upon spin-independent scattering, the non-adiabatic torque can be significantly enhanced. Such an enhancement is large for vortex cores compared to transverse domain walls. We also show that the topological spin currents owing in these structures dramatically enhances the non-adiabaticity, an effect unique to non-trivial topological textures (iii) Antiferromagnetic skyrmions: We extend this study to antiferromagnetic skyrmions and show that such an enhanced topological torque also exist in these systems. Even more interestingly, while such a non-adiabatic torque inuences the undesirable transverse velocity of ferromagnetic skyrmions, in antiferromagnetic skyrmions, the topological non-adiabatic torque directly determines the longitudinal velocity. As a consequence, scaling down the antiferromagnetic skyrmion results in a much more efficient spin torque.