It is well-known that in the absence of sources Maxwell equations are invariant with respect to exchange of electric and magnetic fields. It has been conjectured that a similar duality symmetry
might exist in quantum nonabelian gauge theories, and that confinement of color can be understood as the dual of the Higgs effect. By now there is overwhelming evidence that electric-magnetic duality is realized in certain nonabelian gauge theories with extended supersymmetry. In this talk I will review
the duality conjecture for nonabelian gauge theories and describe some recent developments which also have an impact on mathematics.