In this talk I illustrate the approach that we recently proposed to perform Monte Carlo simulations of lattice quantum field theories (QFTs) with a sign problem. The approach consists in formulating the theory on a "Lefschetz thimble" (that I will introduce). The new formulation does not coincide with any of the usual ones exactly, but I argue, relying on universality, that it is also a legitimate regularization. I then describe the algorithm that we are using to sample the configurations on the thimble. The implementation of the algorithm requires a mapping of the original QFT into a system with one additional dimension. This poses some challenges, but the expected scaling properties are acceptable. Finally, I show the results of our first numerical tests on small lattices for a simple but nontrivial QFT. These are very encouraging in view the wide applicability - in principle - of our approach.