We present an investigation of the X-ray properties of 13 LINERs based on ROSAT all-sky survey and pointed PSPC and HRI observations . Several sources are studied for the first time in X-rays . The X-ray spectra are best described by a powerlaw with photon index \Gamma _ { x } \approx –2.0 or thermal emission from gas with very subsolar abundances . The luminosities range between \log L _ { x } = 37.7 ( NGC 404 ) and 40.8 ( NGC 4450 ) . No X-ray variability on the timescale of hours/days is detected . This is in line with the suggestion that LINERs may accrete in the advection-dominated mode . On a longer term , one of the objects , NGC 2768 , turns out to be slightly variable . Some sources appear to be extended at weak emission levels whereas the bulk of the X-ray emission is consistent with arising from a point source within the PSPC instrumental resolution . L _ { x } / L _ { B } ratios are derived and emission mechanisms that potentially contribute to the observed X-ray luminosities are discussed . We also examine the presence of second X-ray sources near the target sources both in terms of instrumental effects and in terms of ‘ real ’ sources within the LINER galaxies .