  This paper presents new Chandra and BeppoSAX observations aimed at
investigating the optical/X-ray mismatch in the enigmatic class of the
Composite galaxies, discovered by a cross-correlation of IRAS and ROSAT all sky
survey catalogues. These galaxies have been classified as star-forming objects
on the basis of their optical spectra, while the detection of weak broad wings
in the H(alpha) emission in a few of them and their high X-ray luminosity in
the ROSAT band indicated the presence of an active nucleus. The analysis of
Chandra observations for 4 Composites has revealed nuclear point-like sources,
with a typical AGN spectrum (Gamma = 1.7-1.9) and little intrinsic absorption.
A strong flux variability has been observed on different time scales, in
particular most of the sources were brighter at the ROSAT epoch. Although of
relative low luminosity for the AGN class (L(2-10 keV) = 3-60 x 10^{41} erg/s),
the active nucleus is nevertheless dominant in the X-ray domain. At other
wavelengths it appears to be overwhelmed by the starburst and/or host galaxy
light, yielding the Composite classification for these objects.
