Vertex algebras, initially introduced in string theory to describe local operators, have become a fundamental tool in various areas of mathematics. Edward Frenkel's chapter reviews the theory of vertex algebras, emphasizing their algebraic-geometric interpretation and applications. The chapter begins with the axiomatic definition of vertex algebras, which are collections of data including a space of states, a vacuum vector, a shift operator, and a vertex operation. These axioms are motivated by the structure of conformal field theory (CFT) and are equivalent to Borcherds' original definition. The chapter discusses several important properties of vertex algebras and provides examples, such as commutative vertex algebras and the Heisenberg vertex algebra. It explains how to make vertex algebras coordinate-independent by attaching a vector bundle with a flat connection to each conformal vertex algebra. This leads to the definition of conformal blocks and the space of coinvariants, which form a sheaf on the moduli space of curves. Vertex algebras have applications in algebraic geometry, modular functions, and topology. They are particularly useful in the study of moduli spaces and have connections to the geometric Langlands correspondence. The chapter also covers rational vertex algebras, which are particularly relevant to conformal field theory, and discusses orbifolds and the Monster group, including the construction of the Moonshine Module vertex algebra. Finally, the chapter explores the BRST construction and $\mathcal{W}$-algebras, which are obtained from quantum hamiltonian reduction and play a crucial role in the study of conformal field theories and integrable systems.Vertex algebras, initially introduced in string theory to describe local operators, have become a fundamental tool in various areas of mathematics. Edward Frenkel's chapter reviews the theory of vertex algebras, emphasizing their algebraic-geometric interpretation and applications. The chapter begins with the axiomatic definition of vertex algebras, which are collections of data including a space of states, a vacuum vector, a shift operator, and a vertex operation. These axioms are motivated by the structure of conformal field theory (CFT) and are equivalent to Borcherds' original definition. The chapter discusses several important properties of vertex algebras and provides examples, such as commutative vertex algebras and the Heisenberg vertex algebra. It explains how to make vertex algebras coordinate-independent by attaching a vector bundle with a flat connection to each conformal vertex algebra. This leads to the definition of conformal blocks and the space of coinvariants, which form a sheaf on the moduli space of curves. Vertex algebras have applications in algebraic geometry, modular functions, and topology. They are particularly useful in the study of moduli spaces and have connections to the geometric Langlands correspondence. The chapter also covers rational vertex algebras, which are particularly relevant to conformal field theory, and discusses orbifolds and the Monster group, including the construction of the Moonshine Module vertex algebra. Finally, the chapter explores the BRST construction and $\mathcal{W}$-algebras, which are obtained from quantum hamiltonian reduction and play a crucial role in the study of conformal field theories and integrable systems.