In these lectures I review the general structure of electric--magnetic
duality rotations in every even space--time dimension. In four dimensions,
which is my main concern, I discuss the general issue of symplectic covariance
and how it relates to the typical geometric structures involved by N=2
supersymmetry, namely Special K\"ahler geometry for the vector multiplets and
either HyperK\"ahler or Quaternionic geometry for the hypermultiplets. I
discuss classical continuous dualities versus non--perturbative discrete
dualities. How the moduli space geometry of an auxiliary dynamical Riemann
surface (or Calabi--Yau threefold) relates to exact space--time dualities is
exemplified in detail for the Seiberg Witten model of an SU(2) gauge theory.Comment: 56 pages, LaTeX, article.sty, espcrc2.sty. Lecture notes at Trieste
Spring School 199
