This volume is based on the Mathematical Advanced Study Semesters course in geometry given at Penn State U. by Katok and his teaching assistant Climenhaga in the Research Experiences for Undergraduates program in the fall semester of 2007. It contains 36 lectures on surfaces, with a focus on the Euler characteristics. It introduces concepts such as the round sphere, flat torus, Möbius strip, Klein Bottle, and elliptic plane, methods of describing surfaces, metric geometry, and fundamental structures such as topology, combinatorial structure, and Riemannian metrics. It is assumed that readers (advanced undergraduates) have knowledge of calculus, some linear algebra, and rudiments of ODE and real analysis. Annotation ©2008 Book News, Inc., Portland, OR (booknews.com)