  In this paper, we construct the Brownian motion of Liouville Quantum Gravity
with central charge $c=1$ (more precisely we restrict to the corresponding free
field theory). Liouville quantum gravity with $c=1$ corresponds to
two-dimensional string theory and is the conjectural scaling limit of large
planar maps weighted with a $O(n=2)$ loop model or a $Q=4$-state Potts model
embedded in a two dimensional surface in a conformal manner.
  Following \cite{GRV1}, we start by constructing the critical LBM from one
fixed point $x\in\mathbb{R}^2$ (or $x\in\S^2$), which amounts to changing the
speed of a standard planar Brownian motion depending on the local behaviour of
the critical Liouville measure $M'(dx)=-X(x)e^{2X(x)}\,dx$ (where $X$ is a
Gaussian Free Field, say on $\mathbb{S}^2$). Extending this construction
simultaneously to all points in $\mathbb{R}^2$ requires a fine analysis of the
potential properties of the measure $M'$. This allows us to construct a strong
Markov process with continuous sample paths living on the support of $M'$,
namely a dense set of Hausdorff dimension $0$. We finally construct the
associated Liouville semigroup, resolvent, Green function, heat kernel and
Dirichlet form.
  In passing, we extend to quite a general setting the construction of the
critical Gaussian multiplicative chaos that was initiated in
\cite{Rnew7,Rnew12} and also establish new capacity estimates for the critical
Gaussian multiplicative chaos.
