The problem of finding eigenvalues and eigenfunctions and studying their behavior plays a crucial role in modern mathematics. More than a century of study by a legion of researchers, including many of the top names in physics as well as mathematics, has led to many important results and groundbreaking applications. "High Precision Methods in Eigenvalue Problems and their Applications" presents a thorough survey of analytical, asymptotic, numerical, and combined methods for solving eigenvalue problems, including those involving complex systems. The authors pay particular attention to a new method of accelerated convergence for solving problems of the Sturm-Liouville type. They consider boundary conditions of the first, second, and third types and present high-precision asymptotic methods for determining eigenvalues and eigenfunctions of higher oscillation modes. The authors describe the basic concepts of the methods and algorithms presented and illustrate their potential by solving meaningful problems. Readers seeking solutions to specific eigenvalue problems will find that rather than taking a common approach based on the Rayleigh-Ritz, Bubnov-Galiorkin, or finite element methods, the methods and algorithms set forth in this book may be much more easily adapted and extended to their purposes.