Explore metric currents, isoperimetric inequalities, and compactness in this lecture by S. Wenger from the University of Fribourg, Switzerland. Delivered as part of the School and Workshop on "Geometric Measure Theory and Optimal Transport," delve into topics such as nonpositive sectional curvature, growth types and functions, metric space theory, and negative curvature. Examine the chrome of hyperbolic spaces and learn how to fill every curve. The presentation covers introductory concepts, outlines key areas, and provides proofs for relevant theorems, offering a comprehensive overview of these advanced mathematical concepts.