This chapter contains the lecture notes of the course ‘Approximation in Geometry’ given in 2023 at the Convex and Discrete Geometry Summer School in Budapest at the Erdős Center, an affiliate of the Rényi Institute of Mathematics. It covers three areas in the broad field of geometric approximation. First, we consider a classical problem in convexity and computer science, the approximation of a convex body in high dimensional Euclidean space by a convex polytope of small complexity. Next, we study quantitative Helly-type results, and finally, an important tool in the field, John’s decomposition of the identity, a linear algebraic notion concerning rank one operators is discussed. The main goal of these notes is to convey some ideas, teach techniques, and hence, exercises are included.