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 em...