A (convex) polytope is the convex hull of finitely many points. Given that the main prerequisite to define and understand a polytope is linear algebra, this topic is suitable for first-year graduate students. This course will give an introduction to the theory of polytopes and their applications to algebraic combinatorics as well as algebraic geometry. During the course we will discuss an important class of polytopes called generalized permutohedra, which has been receiving a lot of attention recently. We will also cover Newton–Okounkov bodies, which allow us to study the geometry of varieties using polytopes.