This book presents a systematic exposition of the theory of convergence in law for semimartingales, which are a broad class of stochastic processes including discrete-time processes, diffusions, Markov processes, and solutions to stochastic differential equations. The theory is closely related to the concept of weak convergence of probability measures on metric spaces. The authors use the stochastic calculus as a powerful tool to study these processes. The book is divided into several chapters, each focusing on different aspects of the theory, including the characteristics of semimartingales, martingale problems, and the convergence of various types of stochastic processes. The main goal is to provide a complete account of the theory of semimartingales and related topics, such as random measures, and to establish limit theorems for these processes. The authors emphasize the importance of the characteristics of semimartingales, which extend the well-known "Lévy-Khintchine triplet" for processes with independent increments. These characteristics play a crucial role in limit theorems. The book discusses various methods for proving convergence of stochastic processes, including the Prokhorov method and the martingale method. The Prokhorov method involves proving tightness of the sequence of processes and convergence of their finite-dimensional distributions. The martingale method, initiated by Stroock and Varadhan, involves convergence of the triplets of characteristics of the processes. The book also includes a detailed discussion of absolute continuity and singularity of measures, as well as contiguity of sequences of measures. These concepts are essential for understanding the statistical applications of the limit theorems presented in the book. The authors also provide a comprehensive treatment of the Skorokhod topology, which is fundamental for the study of convergence of processes. The book is structured into ten chapters, each covering different aspects of the theory of stochastic processes, semimartingales, and their limit theorems. The authors provide a thorough treatment of the subject, including detailed proofs and examples, making it a valuable resource for researchers and students in the field of probability theory and stochastic processes.This book presents a systematic exposition of the theory of convergence in law for semimartingales, which are a broad class of stochastic processes including discrete-time processes, diffusions, Markov processes, and solutions to stochastic differential equations. The theory is closely related to the concept of weak convergence of probability measures on metric spaces. The authors use the stochastic calculus as a powerful tool to study these processes. The book is divided into several chapters, each focusing on different aspects of the theory, including the characteristics of semimartingales, martingale problems, and the convergence of various types of stochastic processes. The main goal is to provide a complete account of the theory of semimartingales and related topics, such as random measures, and to establish limit theorems for these processes. The authors emphasize the importance of the characteristics of semimartingales, which extend the well-known "Lévy-Khintchine triplet" for processes with independent increments. These characteristics play a crucial role in limit theorems. The book discusses various methods for proving convergence of stochastic processes, including the Prokhorov method and the martingale method. The Prokhorov method involves proving tightness of the sequence of processes and convergence of their finite-dimensional distributions. The martingale method, initiated by Stroock and Varadhan, involves convergence of the triplets of characteristics of the processes. The book also includes a detailed discussion of absolute continuity and singularity of measures, as well as contiguity of sequences of measures. These concepts are essential for understanding the statistical applications of the limit theorems presented in the book. The authors also provide a comprehensive treatment of the Skorokhod topology, which is fundamental for the study of convergence of processes. The book is structured into ten chapters, each covering different aspects of the theory of stochastic processes, semimartingales, and their limit theorems. The authors provide a thorough treatment of the subject, including detailed proofs and examples, making it a valuable resource for researchers and students in the field of probability theory and stochastic processes.