Mathematical logic serves as the cornerstone of formal reasoning, providing precise tools for analyzing the structure and validity of arguments. This paper offers a comprehensive exploration of key topics in mathematical logic, spanning from classical propositional and predicate logic to modal logic and non-classical logics. It examines the syntactic and semantic aspects of various logical systems, delves into proof theory and computational complexity, and explores applications in diverse fields such as mathematics, computer science, philosophy, and linguistics. By elucidating the fundamental principles and practical implications of mathematical logic, this paper highlights its pivotal role in advancing knowledge and addressing complex challenges across disciplines.