After his well-known and widely used book on ordinary differential equations, the famous mathematician Vladimir Arnold now turns his attention to partial differential equations in a new textbook. In his uniquely elegant style, he introduces the topic through a geometric and intuitive approach, allowing readers to gain a deeper understanding of the nature of partial differential equations. This book is a must-read for students of mathematics and physics. Like all of Vladimir Arnold's books, it is filled with geometric insights. Arnold illustrates each principle with a figure. The book covers the most fundamental aspects of the field and primarily focuses on the Cauchy problem and the Neumann problem for the classical linear equations of mathematical physics, particularly the Laplace equation and the wave equation.