In this course, we introduce fundamental algebraic & topological tools and their numerous applications to problems in dynamics and control. Throughout, the basic concepts of category theory—categories, universal properties, and functors—are emphasized as unifying tools. After introducing categories, we study basic topology (spaces, maps, products, quotients) and its applications in topological dynamics (flows, maps, recurrence, Conley's fundamental theorem of dynamical systems). We then study basic group theory (groups, subgroups, products, quotients, free groups) and algebraic topology (fundamental group & Seifert-Van-Kampen) with applications to navigation problems in robotics. As time permits, we will perform a deeper study of algebraic topology (homology, exact sequences, degree theory), and provide applications to vector fields & stabilization (hairy ball theorem, Brockett's theorem) and data analysis (persistent homology).