This is the first comprehensive exposition of the application of spherical harmonics to prove geometric results. All the necessary tools from classical theory of spherical harmonics are presented with full proofs. These tools are used to prove geometric inequalities, uniqueness results for projections and intersection by planes or half-spaces, stability results, and characterisations of convex bodies of a particular type, such as rotors in convex polytopes. Results arising from these analytical techniques have proved useful in many applications, particularly those related to stereology. To make the treatment as self-contained as possible the book begins with background material in analysis and the geometry of convex sets. This reference will be welcomed by both pure and applied mathematicians.