We present the results of the first X-ray study of a sample of 16 young radio sources classified as Compact Symmetric Objects (CSOs). We observed six of them for the first time in X-rays using {it Chandra}, re-observed four with the previous {it XMM-Newton} or {it Beppo-SAX} data, and included six other with the archival data. All the sources are nearby, $z<1$ with the age of their radio structures ($<3000$~years) derived from the hotspots advance velocity. Our results show heterogeneous nature of the CSOs indicating a complex environment associated with young radio sources. The sample covers a range in X-ray luminosity, $L_{2-10,rm keV} sim 10^{41}$-$10^{45}$,erg,s$^{-1}$, and intrinsic absorbing column density of $N_H simeq 10^{21}$--10$^{22}$,cm$^{-2}$. In particular, we detected extended X-ray emission in 1718$-$649; a hard photon index of $Gamma simeq 1$ in 2021$+$614 and 1511$+$0518 consistent with either a Compton thick absorber or non-thermal emission from compact radio lobes, and in 0710$+$439 an ionized iron emission line at $E_{rest}=(6.62pm0.04)$,keV and EW $sim 0.15-$1.4,keV, and a decrease by an order of magnitude in the 2-10 keV flux since the 2008 {it XMM-Newton} observation in 1607$+$26. We conclude that our pilot study of CSOs provides a variety of exceptional diagnostics and highlights the importance of deep X-ray observations of large samples of young sources. This is necessary in order to constrain theoretical models for the earliest stage of radio source evolution and study the interactions of young radio sources with the interstellar environment of their host galaxies.
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