Sturmian Theory for Ordinary Differential Equations I. Historical Prologue 1. Introduction 2. Methods Based Upon Variational Principles 3. Historical Comments on Terminology II. Sturmian Theory for Real Linear Homogeneous Second Order Ordinary Differential Equations on a Compact Interval 1. Introduction 2. Preliminary Properties of Solutions of (1.1) 3. The Classical Oscillation and Comparison Theorems of Sturm 4. Related Oscillation and Comparison Theorems 5. Sturmian Differential Systems 6. Polar Coordinate Transformations 7. Transformations for Differential Equations and Systems 8. Variational Properties of Solutions of (1.1) 9. Comparison Theorems 10. Morse Fundamental Quadratic Forms for Conjugate and Focal Points 11. Survey of Recent Literature 12. Topics and Exercises III. Self Adjoint Boundary Problems Associated with Second Order Linear Differential Equations 1. A Canonical Form for Boundary Conditions 2 Extremum Problems for Self Adjoint Systems 3. Comparison Theorems 4. Comments on Recent Literature 5. Topics and Exercises IV. Oscillation Theory on a Non Compact Interval 1. Introduction 2. Integral Criteria for Oscillation and Non Oscillation 3. Principal Solutions 4. Theory of Singular Quadratic Functionals 5. Interrelations Between Oscillation Criteria and Boundary Problems 6. Strong and Conditional Oscillation 7. A Class of Sturmian Problems on a Non Compact Interval 8. Topics and Exercises V. Sturmian Theory for Differential Systems 1. Introduction 2. Special Examples 3. Preliminary Properties of Solutions of (2.5) 4. Associated Riccati Matrix Differential Equations 5. Normality and Abnormality 6. Variational Properties of Solutions of (3.1) 7. Comparison Theorems 8. Morse Fundamental Hermitian Forms 9. Generalized Polar Coordinate Transformations for Matrix Differential Systems 10. Matrix Oscillation Theory 11. Principal Solutions 12. Comments on Systems (3.1) Which are Not Identically Normal 13. Comments on the Literature on Oscillation Theory for Hamiltonian Systems (3.1) 14. Higher Order Differential Equations 15. Topics and Exercises VI. Self Adjoint Boundary Problems 1. Introduction 2. Normality and Abnormality of Boundary Problems 3. Self Adjoint Boundary Problems Associated with (B) 4. Comparison Theorems 5. Treatment of Self Adjoint Boundary Problems by Matrix Oscillation Theory 6. Notes and Comments on the Literature 7. Topics and Exercises VII. A Class of Definite Boundary Problems 1. Introduction 2. Definitely Self Adjoint Boundary Problems 3. Comments on Related Literature 4. Topics and Exercises VIII. Generalizations of Sturmian Theory 1. Introduction 2. Integro Differential Boundary Problems 3. A Class of Generalized Differential Equations 4. Hestenes Quadratic Form Theory in a Hilbert Space 5. The Weinstein Method of Intermediate Problems 6. Oscillation Phenomena for Hamiltonian Systems in a B* Algebra 7. Topological Interpretations of the Sturmian Theorems Abbreviations for Mathematical Publications Most Frequently Used Special Symbols Author Index.