Several well-established geometric and topological methods are used in this work on a beautiful and important physical phenomenon known as the 'geometric phase.' Going back to the intense interest in this subject since the mid-1980s and the seminal work of M. Berry and B. Simon, this book examines geometric phases, bringing together different physical phenomena under a unified mathematical scheme. Key background material, beginning with the notion of manifolds and differential forms, as well as basic mathematical tools — fiber bundles, connections and holonomies — are presented in Chapter 1. Topological invariants such as Chern classes and homotopy theory are explained in simple, concrete language with emphasis on physical applications. The exposition then unfolds systematically. The adiabatic phases of Berry, the Wilczek — Zee nonabelian factor, and a classical counterpart called Hannay's angles focus on the physical side of the geometric phase problem. Thereafter the geometry of quantum evolution is treated. Here the reader learns about different geometries (such as symplectic and metric structures) living on a quantum phase space in connection with both abelian and nonabelian geometric phases. The concluding section on Examples and Applications paves the way for a continuing study of the geometric PHASES. Throughout the text, material is presented on a level suitable for graduate students and researchers in applied mathematics and physics with an understanding of classical and quantum mechanics.