Riccati equations are ubiquitous in systems/control theory and are frequently solved by the methods of continuous optimization. In some cases, it is known that solutions can be obtained quickly and efficiently by convex-optimization methods, but small modifications to these settings can easily destroy convexity, limiting the applicability of convex-optimization methods. This thesis considers manifold and stochastic optimization techniques to address cases where Riccati-Equations/Riccati Constraints cannot be addressed by convex-optimization methods. Specifically, this thesis studies system-identification, where Riccati constraints are required to ensure synthesizability, and in control synthesis, where Riccati constraints ensure stabilization. The methods developed here yield tractable procedures for identification and synthesis, extending the reach of Riccati-based design beyond what is accommodated by convex-optimization frameworks.