Preface. Acknowledgments. **Special Functions Of Preface. Acknowledgements. Special Functions of the Fractional Calculus**. Gamma Function. Mittag-Leffler Function. Wright Function. **Fractional Derivatives and Integrals**. The Name of the Game. Grünwald-Letnikov Fractional Derivatives. Riemann-Liouville Fractional Derivatives. Some Other Approaches. Sequential Fractional Derivatives. Left and Right Fractional Derivatives. Properties of Fractional Derivatives. Laplace Transforms of Fractional Derivatives. Fourier Transforms of Fractional Derivatives. Mellin Transforms of Fractional Derivatives. **Existence and Uniqueness Theorems**. Linear Fractional Differential Equations. Fractional Differential Equation of a General Form. Existence and Uniqueness Theorem as a Method of Solution. Dependence of a Solution on Initial Conditions. **The Laplace Transform Method**. Standard Fractional Differential Equations. Sequential Fractional Differential Equations. **Fractional Green's Function**. Definition and Some Properties. One-Term Equation. Two-Term Equation. Three-Term Equation. Four-Term Equation. Calculation of Heat Load Intensity Change in Blast Furnace Walls. Finite-Part Integrals and Fractional Derivatives. General Case: *n*-term Equation. **Other Methods for the Solution of Fractional-order Equations**. The Mellin Transform Method. Power Series Method. Babenko's Symbolic Calculus Method. Method of Orthogonal Polynomials. **Numerical Evaluation of Fractional Derivatives**. Approximation of Fractional Derivatives. The "Short-Memory" Principle. Order of Approximation. Computation of Coefficients. Higher-order Approximations. **Numerical Solution of Fractional Differential Equations**. Initial Conditions: Which Problem to Solve? Numerical Solution. Examples of Numerical Solutions. The "Short-Memory" Principle in Initial Value Problems for Fractional Differenti