Abstract: Liouville Brownian motion (LBM) is the natural diffusion process associated to the so-called Liouville measure (formally) defined by the exponential of the Gaussian Free Field, i.e. Gaussian multiplicative chaos with Green kernel. In this talk, I will consider the Liouville heat kernel, defined as the density of the LBM with respect to the Liouville measure. More precisely, I will present regularity estimates and non trivial off-diagonal bounds for the Liouville heat kernel. Based on jointwork with P. Maillard, R. Rhodes and O. Zeitouni.