The differential equations of mathematical physics have a twofold character: their physical content and their mathematical solutions. This book discusses the basic tools of theoretical physicists, applied mathematicians, and engineers, providing detailed insights into linear algebra, Fourier transforms, special functions, Laplace and Poisson, diffusion and vector equations.These basic tools are a set of methods and techniques, known as the equations of mathematical physics. At first sight, they look like a collection of disparate things. Many students in theoretical physics perceive them as strange, autonomous, inflexible, and ultimately unknown objects, whose sole use resides in their being applied to solving usually standard physical problems. While mathematicians are oriented towards empty generalizations and the so-called mathematical rigour, theoretical physicists often limit themselves to giving a set of recipes and examples. Both succeed in producing large, heavy tomes, which are, to a large extent, useless. The only exception seems to be Sommerfeld’s Partielle Differentialgleichungen der Physik, which, however, is rather limited to a restricted list of subjects.The physical nature and origin of the equations of mathematical physics is emphasized in this book, and their various elements and great flexibility are described. The book reveals the indissoluble connection between physical ideas and mathematical concepts, and how these visions can be transcribed into accurate mathematics.