|Résumé||Tensor network states are powerful variational ans¨atze for ground states of quantum many-body systems on a lattice. Monte Carlo sampling techniques have been proposed as a strategy to reduce the computational cost of contractions in tensor network approaches. Here we put forward a variational Monte Carlo approach for the multi-scale entanglement renormalization ansatz (MERA), which is a unitary tensor network. Two major adjustments are required compared to previous proposals with non-unitary tensor networks. First, instead of sampling over configurations of the original lattice, made of L sites, we sample over configurations of an effective lattice, which is made of just log(L) sites. Second, the optimization of unitary tensors must account for their unitary character while being robust to statistical noise, which we accomplish with a modified steepest descent method within the set of unitary tensors. We demonstrate the performance of the variational Monte Carlo MERA approach in the relatively simple context of a finite quantum spin chain at criticality, and discuss future, more challenging applications, including two dimensional systems.