This new volume of the GSM series by the AMS has a much wider scope than the usual textbook on differential geometry. It covers all the basic notions (surfaces in three dimensions and their relations to complex and conformal geometry, smooth manifolds, groups of transformations, including crystallographic groups, tensor algebra and tensor fields, differential forms and their integration, the Stokes theorem and its relation to the de Rham cohomology, connections and their curvature, Riemannian geometry, geodesics and the Gauss-Bonnet formula). The last third of the book treats more advanced topics (conformal geometry, Kähler manifolds, the Morse theory, Hamiltonian formalism, groups of symmetries and conservation laws, Poisson and Lagrange structure on manifolds, and multidimensional calculus of variation). The last chapter describes basic fields of theoretical physics (Einstein’s general relativity, Clifford algebras, spinors and the Dirac equation, and Yang-Mills fields). The authors try to avoid unnecessary abstraction; they present many important topics in modern geometry and topology in a unified way. The book is designed for students in mathematics and theoretical physics but it will be very useful for teachers as well.