An important theme in Mathematics is the interplay of Mathematical structures from seemingly disparate areas. A H-space is such an example: a topological space with a continuous algebraic (multplication) structure: e.g. the unit circle in the complex plane under complex multiplication. A loop space is an even more refined example. Unveiling and studying such structure allows deeper topological and geometric understanding of the underlying space. In recent years, the problem of understanding the topology of spaces of Riemannian Metrics on a smooth manifold M, which satisfy some particular curvature constraint, has attract considerable attention. We discuss some developments in this subject and show how under reasonable conditions, deeper algebraic structures can be uncovered. In particular, such spaces often admit H and Loop space structure.