Introduction -- Black Holes and First Order Flows in Supergravity -- Representations of Super Lie Groups: Some Remarks -- On Chiral Quantum Superspaces -- On the Construction of Chevalley Supergroup -- Indecomposable Finite-dimensional Representations of a Class o f Lie algebras and Lie Superalgebras -- On the Geometry of Super Riemann Surfaces -- Charge Orbits and Moduli Spaces of Black Hole Attractors -- Maximal Supersymmetry -- Lie Supergroups, Unitary Representations, and Invariant Cones -- Geometry of Dual Pairs of Complex Supercurves -- On the Superdimension of an Irreducible Representation.

Supersymmetry was created by the physicists in the 1970's to give a unified treatment of fermions and bosons, the basic constituents of matter. Since then its mathematical structure has been recognized as that of a new development in geometry, and mathematicians have busied themselves with exploring this aspect. This volume collects recent advances in this field, both from a physical and a mathematical point of view, with an accent on a rigorous treatment of the various questions raised.