We present results from a ROSAT Position Sensitive Proportional Counte
r (PSPC) survey of the X-ray emission from Wolf-Rayet (WR) galaxies, a
class of galaxies believed to be young starbursts (with ages of t sim
ilar to 4-6 Myr), many of which are blue compact dwarf galaxies. Of th
e 36 WR galaxies listed in the catalogue of Conti, a total of 14 have
been observed deliberately or serendipitously with the ROSAT PSPC, and
of these, seven have been detected. The derived X-ray luminosities of
WR galaxies range over nearly three orders of magnitude, from L-X sim
ilar to 4 x 10(38) S-1 to 2 x 10(41) erg s(-1). The X-ray spectra of t
he WR galaxies can typically be well-fitted with a single temperature
Raymond-Smith spectral model, with a temperature in the range kT=0.3-1
.0 keV, with the general trend that the more X-ray-luminous WR galaxie
s have hotter spectra. WR galaxies are significantly X-ray-overluminou
s for their blue luminosity, compared with a sample of nearby spiral a
nd starburst galaxies. In addition, the X-ray luminosity of WR galaxie
s correlates well with the far-infrared luminosity L-FIR and the numbe
r of Lyman continuum photons N-Lyc. No strong correlation was found wi
th the equivalent width of the WR emission feature around lambda 4686
Angstrom, the presence of which essentially defines the class of galax
ies. There is little evidence of extended X-ray emission. Various expl
anations for the observed properties of WR galaxies are explored, and
we conclude that the X-ray emission provides strong evidence that a la
rge fraction of the observed X-rays are coming from a hot superbubble
formed by the combined action of stellar winds from massive early-type
stars in the central starburst cluster. These results are consistent
with, and add weight to, the view that WR galaxies are young starburst
s, in which the duration of the star-forming epoch was very short, and
that we are viewing them a few Myr after the initiation of the starbu
rst. As such, WR galaxies represent an important epoch in the evolutio
n of starburst galaxies.