  We present radial velocity (RV) measurements of our sample of 40 M dwarfs
from our planet search programme with VLT+UVES begun in 2000. Although with our
RV precision down to 2 - 2.5 m/s and timebase line of up to 7 years, we are
capable of finding planets of a few Earth masses in the close-in habitable
zones of M dwarfs, there is no detection of a planetary companion. To
demonstrate this we present mass detection limits allowing us to exclude
Jupiter-mass planets up to 1 AU for most of our sample stars. We identified 6 M
dwarfs that host a brown dwarf or low-mass stellar companion. With the
exception of these, all other sample stars show low RV variability with an rms
< 20 m/s. Some high proper motion stars exhibit a linear RV trend consistent
with their secular acceleration. Furthermore, we examine our data sets for a
possible correlation between RVs and stellar activity as seen in variations of
the Halpha line strength. For Barnard's star we found a significant
anticorrelation, but most of the sample stars do not show such a correlation.
