The present thesis consists of two main studies, in the first part, after giving a brief formulation of gravity theories on the metric, affine and metric-affine frameworks, we study the effects of the disformal coupling term \epsilon R_{\mu\nu} V^mu V^nu. We track the effects of disformal term up to field equations, then construct the Einstein tensor G_{\mu\nu} and subsequently identify an effective energy-momentum tensor T_{\mu\nu} to extract effective energy density and pressure. We conclude the first part by comparing the results of metric-affine disformal theory with metrical disformal theory. In the second part, we study the cosmological effects of cross-curvature theory with the functional F (R,R). We derive the both Friedmann equations with the general functional F (R, R) and compare our findings with the known F (R) theory results.