The paper by Mikhael Gromov explores the geometry of Carnot-Caratheodory (C-C) spaces, focusing on their internal structure and properties. It begins with basic definitions and examples, introducing the concept of C-C metrics derived from horizontal curves in a manifold. The paper discusses the connectivity of these spaces, the role of self-similarity, and the relationship between C-C metrics and Riemannian metrics. It also examines the Hausdorff dimension of C-C spaces and the isoperimetric properties of these spaces. The paper then delves into the Heisenberg group as a key example of a C-C space, highlighting its self-similar properties and the implications for the geometry of such spaces. It discusses the use of self-similarity in understanding the asymptotic behavior of C-C spaces and their tangent cones. The paper also addresses the problem of Hölder exponents for maps between C-C spaces, the shape of C-C balls, and the homotopy properties of Hölder maps. Gromov also explores the connection between C-C geometry and hyperbolic geometry, as well as the role of Pansu convergence in the study of nilpotent Lie groups. The paper provides a detailed analysis of the structure of C-C spaces, their invariants, and their relationships with other geometric structures. It concludes with a discussion of the broader implications of these results for the study of non-smooth and singular spaces. The paper is a comprehensive exploration of the internal geometry of C-C spaces, offering new insights into their structure and properties.The paper by Mikhael Gromov explores the geometry of Carnot-Caratheodory (C-C) spaces, focusing on their internal structure and properties. It begins with basic definitions and examples, introducing the concept of C-C metrics derived from horizontal curves in a manifold. The paper discusses the connectivity of these spaces, the role of self-similarity, and the relationship between C-C metrics and Riemannian metrics. It also examines the Hausdorff dimension of C-C spaces and the isoperimetric properties of these spaces. The paper then delves into the Heisenberg group as a key example of a C-C space, highlighting its self-similar properties and the implications for the geometry of such spaces. It discusses the use of self-similarity in understanding the asymptotic behavior of C-C spaces and their tangent cones. The paper also addresses the problem of Hölder exponents for maps between C-C spaces, the shape of C-C balls, and the homotopy properties of Hölder maps. Gromov also explores the connection between C-C geometry and hyperbolic geometry, as well as the role of Pansu convergence in the study of nilpotent Lie groups. The paper provides a detailed analysis of the structure of C-C spaces, their invariants, and their relationships with other geometric structures. It concludes with a discussion of the broader implications of these results for the study of non-smooth and singular spaces. The paper is a comprehensive exploration of the internal geometry of C-C spaces, offering new insights into their structure and properties.