Quantum field theories can easily be studied in perturbation theory when they are weakly coupled, but very little is known about them when they are strongly coupled, even though many interesting field theories (including quantum chromodynamics at low energies) are strongly coupled. For some quantum field theories, it has been found that the strongly coupled theory has an alternative description in terms of some different quantum field theory, that is sometimes weakly coupled. In two space-time dimensions this phenomenon has been known for a long time and is reasonably well-understood. In higher dimensions we have a list of examples of this phenomenon, mostly in supersymmetric field theories, but no general understanding of when and how it happens. Can we find dual descriptions for more strongly coupled field theories in three and four space-time dimensions, in particular for more non-supersymmetric theories ? Can we relate the dualities of different theories (including theories in different space-time dimensions) ? Can we understand duality mappings in general, and obtain rules that will tell us when some theory has a dual description, and what it is ?