  The question of the total measure of invariant tori in analytic,
nearly--integrable Hamiltonian systems is considered. In 1985, Arnol'd, Kozlov
and Neishtadt, in the Encyclopaedia of Mathematical Sciences \cite{AKN1}, and
in subsequent editions, conjectured that in $n=2$ degrees of freedom the
measure of the non torus set of general analytic nearly--integrable systems
away from critical points is exponentially small with the size $\e$ of the
perturbation, and that for $n\ge 3$ the measure is, in general, of order $\e$
(rather than $\sqrt\e$ as predicted by classical KAM Theory). In the case of
generic natural Hamiltonian systems, we prove lower bounds on the measure of
primary and secondary invariant tori, which are in agreement, up to a
logarithmic correction, with the above conjectures. The proof is based on a new
{\sl singular} KAM theory, particularly designed to study analytic properties
in neighborhoods of the secular separatrices generated by the perturbation at
simple resonances.
