Explore algebraic structures on polytopes in this AMS Invited Address from the 2018 Joint Mathematics Meetings. Delve into combinatorics through a personal perspective, examining permutations and associations in polytopes. Discover motivating applications in inverting power series, including multiplication and composition. Learn about Hopf monoids, their definitions, and examples in graphs, posets, and matroids. Investigate the antipode of Hopf monoids and generalized permutahedra, including their coproduct and connections to posets, graphs, and matroids. Examine characters of Hopf monoids and their groups for permutahedra and associahedra. Conclude with current directions in polytope algebra, gaining insights from Federico Ardila of San Francisco State University in this comprehensive 58-minute lecture.