This monograph presents a comprehensive treatment of minimal surfaces and functions of bounded variation. It is divided into two parts: parametric and non-parametric minimal surfaces. The first part focuses on the theory of functions of bounded variation (BV) and Caccioppoli sets, discussing their properties, traces, reduced boundaries, and regularity. It includes the proof of the regularity of minimal hypersurfaces almost everywhere, a key result by De Giorgi, and the analysis of singular sets, showing that their dimension cannot exceed n - 8. The second part addresses non-parametric minimal surfaces, starting with the classical Dirichlet problem for the minimal surface equation. It discusses the existence and regularity of solutions, the role of mean curvature, and the necessity of relaxing the boundary conditions to ensure solvability. The book also covers the Bernstein problem, which concerns the characterization of entire solutions of the minimal surface equation. It proves that in dimensions up to 7, the only entire solutions are affine functions, a result extended by Almgren and Simons. The monograph is self-contained, requiring only a basic knowledge of measure theory and elliptic PDEs. It includes appendices with essential results and references. The text is written for mathematicians interested in geometric measure theory and the calculus of variations, providing a thorough foundation for understanding minimal surfaces and their properties in both parametric and non-parametric settings.This monograph presents a comprehensive treatment of minimal surfaces and functions of bounded variation. It is divided into two parts: parametric and non-parametric minimal surfaces. The first part focuses on the theory of functions of bounded variation (BV) and Caccioppoli sets, discussing their properties, traces, reduced boundaries, and regularity. It includes the proof of the regularity of minimal hypersurfaces almost everywhere, a key result by De Giorgi, and the analysis of singular sets, showing that their dimension cannot exceed n - 8. The second part addresses non-parametric minimal surfaces, starting with the classical Dirichlet problem for the minimal surface equation. It discusses the existence and regularity of solutions, the role of mean curvature, and the necessity of relaxing the boundary conditions to ensure solvability. The book also covers the Bernstein problem, which concerns the characterization of entire solutions of the minimal surface equation. It proves that in dimensions up to 7, the only entire solutions are affine functions, a result extended by Almgren and Simons. The monograph is self-contained, requiring only a basic knowledge of measure theory and elliptic PDEs. It includes appendices with essential results and references. The text is written for mathematicians interested in geometric measure theory and the calculus of variations, providing a thorough foundation for understanding minimal surfaces and their properties in both parametric and non-parametric settings.