The book gives a modern interpretation of the problem of two bodies (two attracting according to the law of universal gravitation of material points), which is the foundation of celestial mechanics and the theory of space flight. In various reference systems, differential equations of motion are derived, phase space is determined, a complete system of integrals and a general solution are built. All degenerate cases are described together with their surroundings. In particular, a unified description of the neighborhood of parabolic movement, as well as the neighborhood of rectilinear movement of any type is given. A detailed presentation of the solution as an explicit function of time in the form of series of three types: according to the degrees of time, according to the degrees of eccentricity, Fourier series . Due attention is paid to convergence issues. The metric spaces of Kepler orbits are introduced and their properties useful in solving the problems of identifying space objects and studying collisions or close passages of celestial bodies are investigated. The main tasks of determining undisturbed orbits are solved. Some of the results were obtained at the Department of Celestial Mechanics C. St Petersburg University . The presentation is accompanied by numerous examples and tasks. The latter are provided with answers, so that the book can serve as a guide. For students and postgraduates of astronomical and astronomical geodetic departments and departments of universities, as well as specialists in the field of celestial mechanics, space geodesy, gravimetry, theory of space flight The electronic version of the publication is posted on the website "Astronomy at St. Petersburg University" (http://www.) astro. spbu. ru)