  The ``complex Plateau problem'' (or boundary problem) in a complexe manifold
X is the problem of characterizing the real submanifolds $\Gamma$ of X which
are boundaries of analytic sub-varieties of $X \backslash \Gamma$. Our
principal result treat the case $X=U\times\omega$ where $U$ is a connected
complex manifold and $\omega$ is a disk-convex K\"ahler manifold. As a
consequence, we obtain results of Harvey-Lawson, Dolbeault-Henkin and Dinh. We
give generalizations of Hartogs-Levi and Hartogs-Bochner theorems. Finally, we
prove that a strictly pseudo-convex CR structure embeddable in a compact
K\"ahler manifold is embeddable in $\CC^n$ if and only if it has a non constant
CR function.
