We will discuss various geometric inequalities motivated by famous existence theorems of various minimal objects in differential geometry proven by topological methods. Let M be a closed Riemannian manifold. Quantitative versions of such theorems as the existence of a periodic geodesic on M due to A. Fet and L. Lusternik, the existence of infinitely many geodesics between an arbitrary pair of points on M (J. P. Serve) and the existence of three simple closed geodesics ona Riemannian 2-sphere (L. Lusternik and L. Schnirelmann) will be presented. We will begin with a discussion of surfaces, next explore how the results for surfaces can be generalized to curvature-free estimates on higher dimensional manifolds. We will next discuss geometric inequalities that involve curvature bounds. If time permits, we will also talk about the case of non-compact complete manifolds with some geometric constraints, like finite volume. In terms of the prerequisites, in addition to Do Carmo's Riemannian Geometry, I would expect students to know some fundamentals of Algebraic Topology, such as Homology and Homotopy Groups, which can be found in Hatcher's textbook