  It has been shown by Paczy\'nski that gravitational microlensing is a useful
method to detect brown dwarfs in the dark halo of our galaxy. At present
several experiments are carried out to monitor several million stars in the
Large Magellanic Cloud with the aim to find such microlensing events. Here I
discuss the possibility to use as targets stars in the Andromeda galaxy. Such
an experiment would be sensitive to both the brown dwarfs in our halo and in
the one of M31. The optical depth $\tau$ to gravitational microlensing due to
brown dwarfs in the halo of M31 turns out to be $\sim 10^{-6}$, which is
comparable to the value of $\tau$ for our own halo. I also compute the
microlensing rate and the average lensing duration and consider moreover the
dependence of $\tau$ on the shape of the halo.
