We dedicate this book to the memory of J. Frank Adams. His clear insights have inspired many mathematicians, including both of us.In January 1989, when the first draft of our book had been completed, we heard the sad news of his untimely death.This has cast a shadow on our subsequent work. Our views of topos theory, as presented here, have been shaped by continued study, by conferences, and by many personal contacts with friends and colleagues-including especially O.Bruno, P. Freyd, J.M.E. Hyland, P.T. Johnstone, A. Joyal, A. Kock, F.W. Lawvere, G.E. Reyes, R Solovay, R Swan, RW. Thomason, M. Tierney, and G.C. Wraith. Our presentation combines ideas and results from these people and from many others, but we have not endeavored to specify the various original sources.Moreover, a number of people have assisted in our work by pro­ viding helpful comments on portions of the manuscript.In this respect, we extend our hearty thanks in particular to P.Corazza, K. Edwards, J. Greenlees, G. Janelidze, G. Lewis, and S. Schanuel.