A striking feature in the study of Riemannian manifolds of positive sectional curvature is the narrowness of the collection of known examples. In this thesis, we examine the structure of the cohomology rings of three families of simply connected seven dimensional Riemannian manifolds that may contain new examples of positive curvature. An explicit computation of these rings reveals that there are infinitely many homotopy types represented in each family. In addition, it becomes possible to identify those manifolds to which there are associated well-known topological invariants distinguishing homeomorphism and diffeomorphism types. All of these manifolds support an action by $S^3 x S^3$ with orbit space a closed interval. Such manifolds are known to be diffeomorphic to the union of the total spaces of two disk bundles. This structure is exploited in two long exact cohomology sequences, which relate the cohomology of the manifold to that of the orbits of the $S^3 x S^3$-action. These sequences, and lemmas derived from them, comprise the primary tools employed in computing the cohomology rings.