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This book provides a coherent introduction to Gutzwiller's trace fromula accessible to well-prepared science, mathematics, and engineering students who have taken introductory courses in linear algebra, calculus, classical, and quantam mechanics. In addition to providing an enrichment of the undergraduate curriculum, this book may serve as the primary text for graduate courses on semiclassical methods. Since periodic-orbit expansions may be used to solve all types of wave systems that typically occur in mathematics, physics, and engineering, this book is attractive for professional scientists and engineers as well. Following a thorough review of elementary concepts in classical and quantum mechanics, the reader is introduced to the idea of classical periodic orbits, the foundation of Gutzwiller's approach to quantum spectra. The trace formual itself is derived following an introduction to Feynman's Path Integrals. Numerous applications, including the exact solutions of "unsolvable' one-dimensional quantum problems, illustrate the power of Gutzwiller's method. Worked examples throughout the text issustrate the material adn provide immediate hadns-on demonstrations of tools and concepts. Key Features: Simultaneously develops both classical adn quantum mechanics, emphasizing the classical-quantum connection Contains introductory chapters on classical chaos and quantum mechanics Includes a section on Feynman's Path Integrals accessible to undergraduate students Contains and application chapter on exact periodic-orbit solutions of quantum graphs, the standard model of quantum complexity Places emphasis on cases where Gutzwiller's trace formula yields exact results Includes exercises at the end of each section and chapter to test the students' mastery of hte material Table of Contents 1. Math Tools 2. Special Functions 3. Newton's Particle Mechancis 4. Schrodinger's Wave Mechanics 5. Semiclassics 6. Test Potentials 7. Periodic-Orbit Quantization 8. Quantum Graphs 9. Advanced Mathematical Techniques 10. Analytical Mechanics 11. Chaos 12. Advanced Cocepts in Quantum Mechanics 13. Semiclassical Propagators 14. Applications Index