An authoritative, self-contained reference text on theoretical and computational seismology Over the past several decades, computational advances have revolutionized seismology, making it possible to simulate seismic wave propagation in complex Earth models and create detailed images of the planet's interior. This cutting-edge text introduces students and scholars to the fundamentals, techniques, and applications of this exciting field of research and discovery. After establishing a strong foundation in continuum mechanics, the book presents the fundamentals of theoretical seismology, providing a basis for subsequent forward and inverse modeling grounded in numerical methods, and then focuses on computational seismology, investigating numerical solutions to seismic wave equations. The adjoint-state method is covered next, along with applications of this technique to waveform inversions across scales, after which the book concludes with a set of appendixes that provide a primer to differential geometry and tensor calculus, which are used throughout the book to explain the fundamental concepts of deformation, strain, and stress from both Eulerian and Lagrangian perspectives. Including over 150 student-tested exercises, the book is an essential resource for motivated students and scholars seeking to master the state of the art of theoretical and computational seismology. Establishes a strong foundation through a geometric analysis of continuum mechanics Reveals how linearizing the resulting equations of motion enables the simulation of seismic wave propagation across nine decades of frequencies and wavelengths Demonstrates how to leverage the capabilities of simulations to create detailed tomographic images from the information embedded in seismographic recordings Covers diverse application areas, including seismology, helioseismology, underwater acoustics, medical imaging, and non-destructive testing Features a wealth of exercises (with online solutions) Includes a comprehensive set of appendixes on differential geometry and tensor calculus An ideal textbook for graduate students studying theoretical seismology, computational seismology, or optimization and inverse problems An essential reference for researchers and scholars.