A majority of disk galaxies host stellar bars that regulate and amplify the flow of angular momentum, J, between disks and their parent dark matter (DM) halos. These bars constitute the prime factor driving internal galaxy evolution. Yet, a non-negligible fraction of disks lack this morphological feature, which led to adoption of the Hubble Fork Diagram. The complex evolution of barred galaxies has been studied by means of numerical simulations, complemented by observations. Despite prolonged efforts, many fundamental questions remain, in part because cosmological simulations still lack the necessary resolution to account for resonant interactions and simulations of isolated galaxies have ignored the cosmological spin of halos. The goal of my thesis is to analyze the J-redistribution in barred galaxies embedded in spinning DM halos, and quantify the DM response. Using high-resolution N-body stellar and DM numerical simulations, I model and analyze the dynamical and secular evolution of stellar bars in disk galaxies and their DM counterparts —induced DM bars in spinning halos with a range of cosmological spin parameter λ ~ 0-0.09. Using a novel method to create initial conditions for the self-consistent equilibrium disk-halo systems, and evolving them for 10 Gyr, I follow the basic parameters of stellar and DM bars, including their observational corollaries. My conclusions are based on nonlinear orbit analysis which quantifies the orbit trapping by the resonances. My main results emphasize a new effect: the DM halo spin has a profound effect on the evolution of stellar and DM bars. Specifically, with increasing λ in the prograde direction: (1) stellar bars develop faster dynamically, but (2) experience a reduced growth during the secular phase of evolution, and basically dissolve for λ > 0.06. These disks can represent the unbarred branch of galaxies on the Hubble Fork Diagram; (3) the stellar bar pattern speeds level off and lose less J; and (4) the stellar bars exhibit ratios of corotation-to-bar radii, RCR/Rbar > 2, representing the so-called slow bars without offset dust lanes. Furthermore, I find that (5) the induced DM bars reach maximal amplitudes which strongly depend on λ, while those of the stellar bars do not; (6) efficiency of resonance trapping of DM orbits by the DM bars, their masses and volumes — all increase with λ; (7) contribution of resonant transfer of J to the DM halo increases with λ as well. (8) prograde and retrograde DM orbits play different roles in J-transfer. (9) Finally, I find that dependence of DM response on λ has important implications for a direct detection of DM and of the associated stellar tracers, such as 'streamers.' Additional results relate the above analysis of corotating disks and halos with those of the counter-rotating ones.