This thesis is devoted to study some interesting cosmic issues in the context of modi- ¯ed Gauss-Bonnet theories. Firstly, we explore the instability ranges of a spherically symmetric anisotropic collapsing °uid under expansion-free condition in f(G) grav- ity. We apply the ¯rst order perturbation scheme to the metric components as well as °uid variables and construct the corresponding ¯eld equations for both static as well as perturbed con¯gurations using viable power-law f(G) model. We establish dynamical equations using contracted Bianchi identities to discuss the dynamical in- stability in both Newtonian and post-Newtonian regimes. It is found that instability ranges depend on energy density, anisotropic pressures and Gauss-Bonnet terms but independent of adiabatic index for expansion-free collapsing °uid. Secondly, we generalize f(G) gravity by introducing non-minimal coupling be- tween Gauss-Bonnet invariant and trace of the energy-momentum tensor named as f(G; T) gravity and explore energy conditions for two reconstructed models in the background of homogeneous and isotropic universe. It is found that the massive test particles move along geodesic trajectories due to the presence of extra force originated from non-zero divergence of the energy-momentum tensor. The energy bounds are expressed in terms of deceleration, jerk and snap cosmological parameters. We study energy conditions for reconstructed models corresponding to de Sitter and power-law cosmological background using pressureless °uid and obtain feasible constraints on free parameters. Thirdly, we discuss stability of the Einstein static universe against homogeneous as well as inhomogeneous scalar perturbations in f(G; T) gravity. We investigate sta- bility regions for particular f(G; T) models corresponding to zero as well as non-zero covariant divergence of the energy-momentum tensor. The graphical analysis shows that stable Einstein universe exists for both spatially closed as well as open universe xi xii models against homogeneous and inhomogeneous perturbations for appropriate choice of parameters. Finally, we analyze stability of some cosmic evolutionary models against linear per- turbations in Hubble parameter and energy density of matter distribution in f(G; T) gravity. We establish the ¯eld equations for both general and particular f(G; T) forms in the context of FRW universe model. We apply the reconstruction technique and found that this theory describes the de Sitter universe, power-law solutions as well as phantom/non-phantom eras cosmological backgrounds. We also discuss stability of de Sitter and power-law reconstructed f(G; T) models and ¯nd stable results against linear perturbations.