Sturm’s 1836 Oscillation Results Evolution of the Theory -- Sturm Oscillation and Comparison Theorems -- Charles Sturm and the Development of Sturm-Liouville Theory in the Years 1900 to 1950 -- Spectral Theory of Sturm-Liouville Operators Approximation by Regular Problems -- Spectral Theory of Sturm-Liouville Operators on Infinite Intervals: A Review of Recent Developments -- Asymptotic Methods in the Spectral Analysis of Sturm-Liouville Operators -- The Titchmarsh-Weyl Eigenfunction Expansion Theorem for Sturm-Liouville Differential Equations -- Sturm’s Theorems on Zero Sets in Nonlinear Parabolic Equations -- A Survey of Nonlinear Sturm-Liouville Equations -- Boundary Conditions and Spectra of Sturm-Liouville Operators -- Uniqueness of the Matrix Sturm-Liouville Equation given a Part of the Monodromy Matrix, and Borg Type Results -- A Catalogue of Sturm-Liouville Differential Equations.This is a collection of survey articles based on lectures presented at a colloquium and workshop in Geneva in 2003 to commemorate the 200th anniversary of the birth of Charles François Sturm. It aims at giving an overview of the development of Sturm-Liouville theory from its historical roots to present day research. It is the first time that such a comprehensive survey is made available in compact form. The contributions come from internationally renowned experts and cover a wide range of developments of the theory. The book can therefore serve both as an introduction to Sturm-Liouville theory and as background for ongoing research. The text is particularly strong on the spectral theory of Sturm-Liouville equations, which has given rise to a major branch of modern analysis. Among other current aspects of the theory discussed are oscillation theory for differential equations and Jacobi matrices, approximation of singular boundary value problems by regular ones, applications to systems of differential equations, extension of the theory to partial differential equations and to non-linear problems, and various generalizations of Borg's inverse theory. A unique feature of the book is a comprehensive catalogue of Sturm-Liouville differential equations covering more than fifty examples, together with their spectral properties. Many of these examples are connected with special functions and with problems in mathematical physics and applied mathematics. The volume is addressed to researchers in related areas, to advanced students and to those interested in the historical development of mathematics. The book will also be of interest to those involved in applications of the theory to diverse areas such as engineering, fluid dynamics and computational spectral analysis.