  Neutral hydrogen clouds are found in the Milky Way and Andromeda halo both as
large complexes and smaller isolated clouds. Here we present a search for Hi
clouds in the halo of M33, the third spiral galaxy of the Local Group. We have
used two complementary data sets: a 3^o x 3^o map of the area provided by the
Arecibo Legacy Fast ALFA (ALFALFA) survey and deeper pointed observations
carried out with the Arecibo telescope in two fields that permit sampling of
the north eastern and south-western edges of the HI disc. The total amount of
Hi around M33 detected by our survey is $\sim 10^7$ M$_{\odot}$. At least 50%
of this mass is made of HI clouds that are related both in space and velocity
to the galaxy. We discuss several scenarios for the origin of these clouds
focusing on the two most interesting ones: $(a)$ dark-matter dominated gaseous
satellites, $(b)$ debris from filaments flowing into M33 from the intergalactic
medium or generated by a previous interaction with M31. Both scenarios seem to
fit with the observed cloud properties. Some structures are found at anomalous
velocities, particularly an extended HI complex previously detected by Thilker
et al. (2002). Even though the ALFALFA observations seem to indicate that this
cloud is possibly connected to M33 by a faint gas bridge, we cannot firmly
establish its extragalactic nature or its relation to M33. Taking into account
that the clouds associated with M33 are likely to be highly ionised by the
extragalactic UV radiation, we predict that the total gas mass associated with
them is > 5 x 10^7 M$_{\odot}$. If the gas is steadily falling towards the M33
disc it can provide the fuel needed to sustain a current star formation rate of
0.5 M$_{\odot}$ yr$^{-1}$.
