The presented book is, as are all V. Arnold's books, full of geometric insight. Its aim is to cover most basic parts of the subject, in particular the Cauchy and Neumann problems for the classical linear equations of mathematical physics. In the preface to the second Russian edition (which is included in the book) one reads: “The author...has attempted to make students of mathematics with minimal knowledge...acquainted with a kaleidoscope of fundamental ideas of mathematics and physics. Instead of the principle of maximal generality, which is usual in mathematical books, the author has attempted to adhere to the principle of minimal generality, according to which every idea should first be clearly understood in the simplest situation; only then can the developed method be extended to more complicated cases....” The book follows a series of lectures, which were delivered to the third year students in the Mathematical College of the Independent University of Moscow during the fall semester of 1994/95. The examination problems at the end of the book form an essential part of it.