Here is the abstract of a paper that is now available in Postscript form at: http://math.ucr.edu/home/baez/ncat.ps If downloading it or printing it out is a problem, I can mail copies to people. ---------------------------------------------------------------------- An Introduction to n-Categories John C. Baez An n-category is some sort of algebraic structure consisting of objects, morphisms between objects, 2-morphisms between morphisms, and so on up to n-morphisms, together with various ways of composing them. We survey various concepts of n-category, with an emphasis on `weak' n-categories, in which all rules governing the composition of j-morphisms hold only up to equivalence. (An n-morphism is an equivalence if it is invertible, while a j-morphism for j < n is an equivalence if it is invertible up to a (j+1)-morphism that is an equivalence.) We discuss applications of weak n-categories to various subjects including homotopy theory and topological quantum field theory, and review the definition of weak n-category recently proposed by Dolan and the author.