Advanced Theoretical Mechanics deals with advanced theoretical mechanics in three dimensions, making use of concepts and methods such as matrices, vectors, tensors, and transformation methods. The definition of a vector via the transformation law obeyed by its components is emphasized, and matrix methods are used to handle sets of components. Special attention is given to the definition of angular velocity and the proof that it can be represented by a vector. This book is comprised of 11 chapters and begins with an introduction to kinematics in three dimensions. Lagrange's equations and analytical dynamics are then presented, along with the simpler problems of three-dimensional dynamics, often with the help of rotating axes. Stability and small oscillations are also considered. The subsequent chapters focus on the dynamics of a particle and the motion of a system of particles; gyroscopic motion, free rotation, and steady motion; oscillations of a dynamical system with a finite number of degrees of freedom; and the vibrations of strings. The final chapter is devoted to analytical dynamics, paying particular attention to Hamilton's principle and equations of motion as well as the Hamilton-Jacobi equation. This monograph is intended for engineers and scientists as well as students of mathematics, physics, and engineering.