We exploit some common features of black hole and domain wall solutions of (super)gravity theories coupled to scalar fields and construct a class of stable extremal black holes that are non-BPS, but still can be described by first-order differential equations. These are driven by a "superpotential", which replaces the central charge Z in the usual black hole potential. We provide a general procedure for finding this class and deriving the associated "superpotential". We also identify some other cases which do not belong to this class, but show a similar behaviour.

We give the full lagrangean and supersymmetry transformation rules for D = 5, N = 2 supergravity interacting with an arbitrary number of vector, tensor and hyper-multiplets, with gauging of the R-symmetry group SU (2) R as well as a subgroup K of the isometries of the scalar manifold. Among the many possible applications, this theory provides the setting where a supersymmetric brane-world scenario could occur. We comment on the presence of AdS vacua and BPS solutions that would be relevant towards a supersymmetric smooth realization of the Randall-Sundrum "alternative to compactification". We also add some remarks on the connection between this most general 5D fully coupled supergravity model and type IIB theory on the T 11 manifold.

We consider generic features of eleven dimensional supergravity compactified down to five dimensions on an arbitrary Calabi-Yau threefold.

We establish general properties of supersymmetric flow equations and of the superpotential of five-dimensional N = 2 gauged supergravity coupled to vector multiplets and hypermultiplets. We provide necessary and sufficient conditions for BPS domain walls and find a set of algebraic attractor equations for N = 2 critical points.As an example we describe in detail the gauging of the universal hypermultiplet and a vector multiplet. We study a two-parameter family of superpotentials with supersymmetric AdS critical points and we find, in particular, an N = 2 embedding for the UV-IR solution of Freedman, Gubser, Pilch and Warner of the N = 8 theory. We comment on the relevance of these results for brane world constructions.

We report on the formulation of N = 2 D = 4 supergravity coupled to n V abelian vector multiplets in presence of electric and magnetic charges. General formulae for the (moduli dependent) electric and magnetic charges for the n V + 1 gauge fields are given which reflect the symplectic structure of the underlying special geometry. The specification to Type IIB strings compactified on Calabi-Yau manifolds, with gauge group U(1) h 21 +1 is given. We report on the formulation of N = 2, D = 4 supergravity coupled to nV abelian vector multiplets in presence of electric and magnetic charges. General formulae for the (moduli dependent) electric and magnetic charges for the nV + 1 gauge fields are given which reflect the symplectic structure of the underlying special geometry. Model independent sum rules obeyed by these charges are obtained. The specification to Type IIB strings compactified on Calabi-Yau manifolds, with gauge group U (1) h 21 +1 , is given. Contribution to the Proceedings of the