This monograph, which is still under development, aims to systematically study analytic curves in the sense of Berkovich over an ultrametric field. It relies on the foundational works of Berkovich and the author's previous research on dimension theory and quasi-smooth spaces. The main focus is on the local and global study of analytic curves, demonstrating that every curve admits a triangulation and deducing the semi-stable reduction theorem. The first chapter introduces the theory of real graphs, covering general concepts, properties of trees and graphs, and the compactification of graphs. It also discusses subgraphs, skeletons, and homotopies, providing a foundation for the subsequent analysis. The second chapter delves into the structure of balls in ultrametric spaces, defining the space of balls $\mathbb{N}(E)$ and the compactification $\mathbb{D}(E)$. It explores the order structure, convexity, and compactness properties of these spaces, as well as the relationship between balls and chains in $\mathbb{D}(E)$. The third chapter examines quotients of graphs, including actions of groups, compactifications, and quotient spaces. It also discusses the algebraic and commutative aspects, such as étale morphisms and Henselian rings, and their applications to analytic geometry. The fourth chapter focuses on local and global studies of analytic curves, including surgery, valuations, and branches. It provides detailed proofs and examples to support the theoretical developments. The fifth chapter introduces triangulations of quasi-smooth analytic curves, detailing the construction and properties of triangulations. It also explores the connection between triangulations and étale cohomology. The sixth chapter discusses formal models of analytic curves in the context of Berkovich geometry, including semi-stable reduction and valuation theory. The monograph is still a work in progress, and the authors acknowledge the need for further references and improvements. Despite this, it has already been cited and is made available on arXiv for broader accessibility.This monograph, which is still under development, aims to systematically study analytic curves in the sense of Berkovich over an ultrametric field. It relies on the foundational works of Berkovich and the author's previous research on dimension theory and quasi-smooth spaces. The main focus is on the local and global study of analytic curves, demonstrating that every curve admits a triangulation and deducing the semi-stable reduction theorem. The first chapter introduces the theory of real graphs, covering general concepts, properties of trees and graphs, and the compactification of graphs. It also discusses subgraphs, skeletons, and homotopies, providing a foundation for the subsequent analysis. The second chapter delves into the structure of balls in ultrametric spaces, defining the space of balls $\mathbb{N}(E)$ and the compactification $\mathbb{D}(E)$. It explores the order structure, convexity, and compactness properties of these spaces, as well as the relationship between balls and chains in $\mathbb{D}(E)$. The third chapter examines quotients of graphs, including actions of groups, compactifications, and quotient spaces. It also discusses the algebraic and commutative aspects, such as étale morphisms and Henselian rings, and their applications to analytic geometry. The fourth chapter focuses on local and global studies of analytic curves, including surgery, valuations, and branches. It provides detailed proofs and examples to support the theoretical developments. The fifth chapter introduces triangulations of quasi-smooth analytic curves, detailing the construction and properties of triangulations. It also explores the connection between triangulations and étale cohomology. The sixth chapter discusses formal models of analytic curves in the context of Berkovich geometry, including semi-stable reduction and valuation theory. The monograph is still a work in progress, and the authors acknowledge the need for further references and improvements. Despite this, it has already been cited and is made available on arXiv for broader accessibility.