In this talk, we study two different geometric flows. First, we introduce the Mean Curvature Flow, an evolution equation for submanifolds of some Euclidean space. The goal is to obtain a good intuition for the formation of singularities along this flow, in particular, we will see many explicit examples and pictures. In the second part of the talk, we discuss the Ricci Flow, which might be seen as the intrinsic analog of the Mean Curvature Flow for abstract Riemannian manifolds. We explain attempts to adopt the results from the first part of the talk to the intrinsic setting and present (partial) results as well as open problems. Some of the proofs of our Ricci Flow results will be presented in the Geometry and Analysis Seminar on Tuesday, October 15.