This lecture lies at the interface of geometry and optimization. In particular, I'll talk about geodesic convexity and geodesically convex optimization. This class of problems offers a rich and multifaceted class of nonconvex optimization problems that can admit tractable global solutions. I'll provide some background on geodesic convexity and some canonical examples as motivation. Subsequently, I will cover complexity theory of gradient based geodesically convex optimization, culminating with the Riemannian analog of Nesterov's Euclidean accelerated gradient method. I will also highlight some open problems and important directions worthy of future pursui