  We construct a stochastic process, called the Liouville Brownian motion,
which is the Brownian motion associated to the metric $e^{\gamma X(z)}\,dz^2$,
$\gamma<\gamma_c=2$ and $X$ is a Gaussian Free Field. Such a process is
conjectured to be related to the scaling limit of random walks on large planar
maps eventually weighted by a model of statistical physics which are embedded
in the Euclidean plane or in the sphere in a conformal manner. The construction
amounts to changing the speed of a standard two-dimensional Brownian motion
$B_t$ depending on the local behavior of the Liouville measure
"$M_{\gamma}(dz)=e^{\gamma X(z)}\,dz$". We prove that the associated Markov
process is a Feller diffusion for all $\gamma<\gamma_c=2$ and that for all
$\gamma<\gamma_c$, the Liouville measure $M_{\gamma}$ is invariant under
$P_{\mathbf{t}}$. This Liouville Brownian motion enables us to introduce a
whole set of tools of stochastic analysis in Liouville quantum gravity, which
will be hopefully useful in analyzing the geometry of Liouville quantum
gravity.
