Vertex algebras, introduced in the context of string theory and later formalized in representation theory, have become central in mathematics. They provide a mathematical framework for two-dimensional conformal field theory, with vertex operators corresponding to chiral symmetry algebras. The key property of associativity in vertex algebras is equivalent to the operator product expansion in conformal field theory. Vertex algebras have found applications in various areas of mathematics, including representation theory of infinite-dimensional Lie algebras, algebraic geometry, finite group theory, modular functions, and topology. Beilinson and Drinfeld introduced chiral algebras, a geometric version of vertex algebras, which have led to new concepts and techniques in algebraic geometry. This talk reviews the theory of vertex algebras with a focus on their algebraic-geometric interpretation and applications. It begins with the axiomatic definition of vertex algebras, which is equivalent to Borcherds' original definition. The text then discusses important properties and examples, including infinite-dimensional Lie algebras such as Heisenberg, affine Kac-Moody, and Virasoro algebras, as well as more unconventional examples like W-algebras. The text explains how to make vertex algebras coordinate independent, leading to the definition of conformal blocks on algebraic curves. These conformal blocks give rise to "chiral correlation functions" on powers of the curve with singularities along the diagonals. The study of these structures has led to insights into the structure of moduli spaces, particularly through the study of D-modules. The text also discusses the relation between vertex algebras and the Beilinson-Drinfeld chiral algebras, which are factorization algebras on the Ran space of finite subsets of a curve. These algebras have been used to construct the conjectural geometric Langlands correspondence between automorphic D-modules and flat L-G-bundles on a curve. The text provides examples of vertex algebras, including the Heisenberg vertex algebra, affine Kac-Moody algebras, the Virasoro algebra, and W-algebras. It also discusses the boson-fermion correspondence, which establishes an isomorphism between a vertex superalgebra and a vertex algebra built from the Heisenberg algebra. Rational vertex algebras, which are particularly relevant to conformal field theory, are discussed, along with their modules and characters. The text also covers orbifolds and the Monster group, as well as the coset construction and BRST construction, which lead to the definition of W-algebras. The Moonshine Module vertex algebra is also discussed, which is a holomorphic vertex algebra with a unique simple module.Vertex algebras, introduced in the context of string theory and later formalized in representation theory, have become central in mathematics. They provide a mathematical framework for two-dimensional conformal field theory, with vertex operators corresponding to chiral symmetry algebras. The key property of associativity in vertex algebras is equivalent to the operator product expansion in conformal field theory. Vertex algebras have found applications in various areas of mathematics, including representation theory of infinite-dimensional Lie algebras, algebraic geometry, finite group theory, modular functions, and topology. Beilinson and Drinfeld introduced chiral algebras, a geometric version of vertex algebras, which have led to new concepts and techniques in algebraic geometry. This talk reviews the theory of vertex algebras with a focus on their algebraic-geometric interpretation and applications. It begins with the axiomatic definition of vertex algebras, which is equivalent to Borcherds' original definition. The text then discusses important properties and examples, including infinite-dimensional Lie algebras such as Heisenberg, affine Kac-Moody, and Virasoro algebras, as well as more unconventional examples like W-algebras. The text explains how to make vertex algebras coordinate independent, leading to the definition of conformal blocks on algebraic curves. These conformal blocks give rise to "chiral correlation functions" on powers of the curve with singularities along the diagonals. The study of these structures has led to insights into the structure of moduli spaces, particularly through the study of D-modules. The text also discusses the relation between vertex algebras and the Beilinson-Drinfeld chiral algebras, which are factorization algebras on the Ran space of finite subsets of a curve. These algebras have been used to construct the conjectural geometric Langlands correspondence between automorphic D-modules and flat L-G-bundles on a curve. The text provides examples of vertex algebras, including the Heisenberg vertex algebra, affine Kac-Moody algebras, the Virasoro algebra, and W-algebras. It also discusses the boson-fermion correspondence, which establishes an isomorphism between a vertex superalgebra and a vertex algebra built from the Heisenberg algebra. Rational vertex algebras, which are particularly relevant to conformal field theory, are discussed, along with their modules and characters. The text also covers orbifolds and the Monster group, as well as the coset construction and BRST construction, which lead to the definition of W-algebras. The Moonshine Module vertex algebra is also discussed, which is a holomorphic vertex algebra with a unique simple module.