This thesis studies some cosmic aspects in modified theories involving cur- vature matter coupling. In this setting, we concentrate on f (R, T ) and f (R, T, R μν T μν ) theories to discuss the thermodynamic laws with the non- equilibrium description at the apparent horizon of FRW universe. It is shown that Friedmann equations can be transformed to the form of Clau- sius relation T h S ef f = δQ, S ef f is the entropy which contains contributions both from horizon entropy as well as additional entropy term introduced due to the non-equilibrating description and δQ is the energy flux across the horizon. The generalized second law of thermodynamics is also estab- lished in a more comprehensive form and one can recover the corresponding results in Einstein as well as f (R) theories. We remark that equilibrium description in such theories needs more study to follow. Moreover, we discuss the validity of energy conditions in f (R, T, R μν T μν ) gravity. The corresponding energy conditions are presented in terms of re- cent values of Hubble, deceleration, jerk and snap parameters. In particular, we use two specific models recently developed in literature to study concrete application of these conditions as well as Dolgov-Kawasaki instability. We explore f (R, T ) gravity as a specific case to this modified theory for expo- nential and power law models. The exact power law solutions are obtained for two particular cases in homogeneous and isotropic f (R, T ) cosmology. Finally, we find certain constraints which have to be satisfied to ensure that power law solutions may be stable and match the bounds prescribed by the energy conditions. We also explore the locally rotationally symmetric Bianchi type I model ixx with perfect fluid as matter content in f (R, T ) gravity. The exact solutions of the field equations are obtained for two expansion laws namely exponen- tial and power law expansions. The physical and kinematical quantities are examined for both cases in future evolution of the universe. We investigate the validity of null energy condition and conclude that our solutions are consistent with the current observations.