  The goal of this work is give a precise numerical description of the K\"ahler
cone of a compact K\"ahler manifold. Our main result states that the K\"ahler
cone depends only on the intersection form of the cohomology ring, the Hodge
structure and the homology classes of analytic cycles: if $X$ is a compact
K\"ahler manifold, the K\"ahler cone $\cK$ of $X$ is one of the connected
components of the set $\cP$ of real $(1,1)$ cohomology classes $\{\alpha\}$
which are numerically positive on analytic cycles, i.e. $\int_Y\alpha^p>0$ for
every irreducible analytic set $Y$ in $X$, \hbox{$p=\dim Y$}. This result is
new even in the case of projective manifolds, where it can be seen as a
generalization of the well-known Nakai-Moishezon criterion, and it also extends
previous results by Campana-Peternell and Eyssidieux. The principal technical
step is to show that every nef class $\{\alpha\}$ which has positive highest
self-intersection number $\int_X\alpha^n>0$ contains a K\"ahler current; this
is done by using the Calabi-Yau theorem and a mass concentration technique for
Monge-Amp\`ere equations. The main result admits a number of variants and
corollaries, including a description of the cone of numerically effective
$(1,1)$ classes and their dual cone. Another important consequence is the fact
that for an arbitrary deformation $\cX\to S$ of compact K\"ahler manifolds, the
K\"ahler cone of a very general fibre $X_t$ is ``independent'' of $t$, i.e.\
invariant by parallel transport under the $(1,1)$-component of the Gauss-Manin
connection.
