This is a survey, at the elementary level, of some of the most important concepts of mathematics. Attention is paid to their technical features, historical development and broader philosophical significance. Each of the various branches of mathematics is discussed separately, but their interdependence is emphasized throughout. Certain topics - such as Greek mathematics, abstract algebra, set theory, geometry and the philosophy of mathematics - are discussed in detail. Appendices outline from scratch the proofs of two of the most celebrated limitative results of mathematics: the insolubility of the problem of doubling the cube and trisecting an arbitrary angle, and the Godel incompleteness theorems. Additional appendices contain brief accounts of smooth infinitesimal analysis - a new approach to the use of infinitesimals in the calculus - and of the philosophical thought of the great 20th-century mathematician Hermann Weyl.