Class field theory describes the Abelian extensions of a local or global field in terms of the arithmetic of the field itself. The aim of this thesis is to present and prove its main statements. We begin by developing local class field theory and then we derive the global results from the local results. An important tool in the proofs, which are strongly influenced by Tate s approach, is Tate cohomology. To bridge the gap between local and global class field theory, we need to introduce idèles. This construction suggests studying the prime ideals of a number field and its complex embeddings in a unified way, which is achieved via valuation theory. Our approach is purely algebraic, avoiding the use of analysis to prove purely algebraic results. Once we get the main statements of global class field theory in terms of idèles, we use them to derive the classical ideal-theoretic formulation. At the end, we give some applications of class field theory.