MPEJ Volume 6, No.2, 18 pp. Received Nov 19 1999, Revised Feb 10 2000, Accepted Feb 15 2000 E. Valdinoci Families of whiskered tori for a-priori stable/unstable Hamiltonian systems and construction of unstable orbits ABSTRACT: We give a detailed statement of a KAM theorem about the conservation of partially hyperbolic tori on a fixed energy level for an analytic Hamiltonian $H(I,\f,p,q)=h(I,pq;\m)+\m f(I,\f,p,q;\m)$, where $\f$ is a $({d}-1)-$dimensional angle, $I$ is in a domain of $\RR^{{d}-1}$, $p$ and $q$ are real in a neighborhood $0$, and $\m$ is a small parameter. We show that invariant whiskered tori covering a large measure exist for sufficiently small perturbations. The associated stable and unstable manifolds also cover a large measure. Moreover, we show that there is a geometric organization to these tori. Roughly, the whiskered tori we construct are organized in smooth families, indexed by a Cantor parameter. The whole set of tori as well as their stable and unstable manifolds is smoothly interpolated. In particular, we emphasize the following items: sharp estimates on the relative measure of the surviving tori on the energy level, analyticity properties, including dependence upon parameters, geometric structures. We apply these results to both ``a-priori unstable'' and ``a-priori stable'' systems. We also show how to use the information obtained in the KAM Theorem we prove to construct unstable orbits.