The research described in this thesis focuses on the investigation of magnetization processes and spin dynamics in the Kitaev magnets. These magnets are described by the generalized Kitaev Hamiltonians including other sub-dominant interactions allowed by symmetry, e.g., nearest neighbor isotropic Heisenberg interaction and the symmetric off-diagonal exchange anisotropy, the so-called \Gamma interaction. Particularly, I concentrate on studying the ground-state phase diagrams of these models and finding possible ground-state spin configurations by using a combination of analytical and numerical methods, such as Luttinger-Tisza method, symmetry analysis, variational and nonlinear energy minimizations. I have shown that subtle interplay of anisotropic bond-dependent magnetic couplings and geometrical frustration leads to rich variety of classical and quantum phases with unusual elementary excitations which strongly depend on the underlying lattice and bonding geometries. I have analyzed the role of quantum fluctuations in these magnetic phases by employing semiclassical spin-wave analysis. I also studied finite temperature properties of these models and the effects of magnetic field, as the presence of anisotropic interactions in these models significantly affects the nature of finite temperature phase transitions and magnetization processes. To perform the studies at finite temperatures, I have used extensive Monte Carlo simulations.