After the appearance of Baker's fundamental papers ‘Linear forms in the logarithms of algebraic numbers’ in Mathematika in 1966–8, Baker, Coates and others obtained upper bounds for the magnitudes of integer solutions of some polynomial diophantine equations in two unknowns and their p-adic generalisations. The finiteness of the numbers of solutions of these equations had been proved by Thue, Siegel, Mahler and others much earlier. The publication of Baker's papers ‘A sharpening of the bounds for linear forms in logarithms’ in Acta Arithmetica in 1972–5, and van der Poorten's p-adic analogues of it, led to completely new results on exponential diophantine equations such as the work on the Catalan equation by Tijdeman and its p-adic analogue by van der Poorten. Since the numerous publications on exponential diophantine equations are scattered over journals and no thorough introduction is available, we have decided to write a tract on these results.

We were together at the University of Leiden in 1982–3 for one year. A first draft of the manuscript was written during this period. The subsequent work of finalising the manuscript was carried out by correspondence spread over a period of about two years. The stay of one of us (T.N.S.) at the University of Leiden was supported in part by the Netherlands Organisation for the Advancement of Pure Research (Z.W.O.).

We are very grateful to K. Györy for his generous help in preparing the manuscript. Lemma A. 16, corollary A.7, theorem 1.4, corollary 1.3, theorem 5.5, theorem 7.2, theorem 7.6 and corollary 7.4 were added or modified on his advice, and he assisted in writing the proofs of these results as well as the changes entailed in other proofs.