Modeling thermodynamic distance curvature and fluctuations : a geometric approach Introduction Algebraic Structures. Spaces. Reference Frames Calculus with Vectors and Differential Forms Elements of Riemann Geometry Equilibrium Thermodynamics Thermodynamics as a Contact Geometry Structure Thermodynamic Distance Geometrization of Thermodynamic Fluctuations Thermodynamic Curvature. Correlation. Stability. This textbook aims to briefly outline the main directions in which the geometrization of thermodynamics has been developed in the last decades. The textbook is accessible to people trained in thermal sciences but not necessarily with solid formation in mathematics. For this in the first chapters a summary of the main mathematical concepts is made. In some sense this makes the textbook self consistent. The rest of the textbook consists of a collection of results previously obtained in this young branch of thermodynamics. The manner of presentation used throughout the textbook is adapted for ease of access of readers with education in natural and technical sciences