Vertex algebras arose out of conformal field theory, and were first defined mathematically by Borcherds in 1986. Since then, they have found applications in many areas of mathematics, including representation theory, number theory, finite group theory, and geometry. Vertex algebras are vector spaces (generally infinite-dimensional) which are equipped with a family of bilinear products (indexed by the integers) which in general are neither commutative nor associative. In many ways they behave like ordinary associative algebras, and the usual categorical and formal algebraic notions like homomorphisms, ideals, quotients, and modules over vertex algebras are easy to define. In this talk, I'll define vertex algebras, give some basic examples, indicate how to do computations, and hopefully state some interesting open problems.