  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-categories recently proposed by Dolan and the author.
