Higher-dimensional paper available, from http://www.dpmms.cam.ac.uk/~leinster. Structures in Higher-Dimensional Category Theory This is an exposition of some of the constructions which have arisen in higher-dimensional category theory. We start with a review of the general theory of operads and multicategories. Using this we give an account of Batanin's definition of n-category; we also give an informal definition in pictures. Next we discuss Gray-categories and their place in coherence problems. Finally, we present various constructions relevant to the opetopic definitions of n-category. New material includes a suggestion for a definition of lax cubical n-category; a characterization of small Gray-categories as the small substructures of 2-Cat; a conjecture on coherence theorems in higher dimensions; a construction of the category of trees and, more generally, of n-pasting diagrams; and an analogue of the Baez-Dolan slicing process in the general theory of operads. (A few corrections have been made to the version of this distributed at the PSSL in Utrecht, and these are listed at the web site.) Tom Leinster