After a short survey of results obtained by Thue and
Siegel, we explain how the theory of linear forms in the logarithms of algebraic numbers, developed by Alan Baker, applies to Diophantine equations and provides us with explicit upper bounds for the size of the solutions of certain families of equations. Unfortunately, these upper bounds are huge, and we cannot hope to list all the solutions by brutal enumeration. However, many important
progress have been accomplished during the last ten years and, by combining various methods, it is now possible to completely solve some exponential Diophantine equations. For instance, in collaboration with Mignotte and Siksek, we have proved that 1, 8 and 144 are the only perfect powers in the Fibonacci sequence.