The book gives a general presentation of some recent developments in wavelet theory, with an emphasis on techniques that are both fundamental and relatively timeless, having a geometric and spectral-theoretic flavour. The exposition is clearly motivated and unfolds systematically, aided by numerous graphics. Excellent graphics show how wavelets depend on the spectra of the transfer operators. Some new results are presented for the first time (e.g., results on homotopy of multiresolutions, on approximation theory, and the spectrum of associated transfer and subdivision operators). The book is divided into six chapters. Each chapter, and some sections within chapters, open with tutorials or primer of varying length. The tutorials are written in a style that is much more informal. They are in fact meant as friendly invitations to the topics to follow, with the emphasis on friendliness. Key topics of wavelet theory are examined: connected components in the variety of wavelets, the geometry of winding numbers, the Galerkin projection method, classical functions of Weierstrass and Hurwitz and their role in describing the eigenvalue-spectrum of the transfer operator, Perron-Frobenius theory, and quadrature mirror filters. Concise background material for each chapter, open problems, exercises, bibliography, and comprehensive index, make this work a fine pedagogical and reference source. The book also describes important applications to signal processing, communications engineering, computer graphics algorithms, qubit algorithms and chaos theory, and is aimed at a broad readership of graduate students, practitioners, and researchers in applied mathematics and engineering. The book is also useful for other mathematicians with an interest in the interface between mathematics and communication theory.