This book is a collection of survey articles on several topics related to the general notion of integrability. It stems from a workshop on 'Mathematical Methods of Regular Dynamics' dedicated to Sophie Kowalevski. Leading experts introduce corresponding areas in depth. The book provides a broad overview of research, from the pioneering work of the nineteenth century to the developments of the 1970s through the present. The book begins with two historical papers by R. L. Cooke on Kowalevski's life and work. Following are 15 research surveys on integrability issues in differential and algebraic geometry, classical complex analysis, discrete mathematics, spinning tops, Painleve equations, global analysis on manifolds, special functions, etc.It concludes with Kowalevski's famous paper published in ""Acta Mathematica"" in 1889, 'Sur le probleme de la rotation d'un corps solide autour d'un point fixe'. The book is suitable for graduate students in pure and applied mathematics, the general mathematical audience studying integrability, and research mathematicians interested in differential and algebraic geometry, analysis, and special functions.