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The Monte Carlo evaluation of path integrals is one of a few general purpose methods to approach strongly coupled systems. It is used in all branches of Physics, from QCD/nuclear physics to the correlated electron systems. However, many systems of great importance (dense matter inside neutron stars, the repulsive Hubbard model away from half-filling, dynamical and non-equilibrium observables) are not amenable to the Monte Carlo method as it currently stands due to the so-called "sign-problem". We review a new set of ideas recently developed to tackle the sign problem based on the complexification of field space and the Picard-Lefshetz theory accompanying it. The mathematical ideas underpinning this approach, as well as the algorithms so far developed, are described together with non-trivial examples where the method has already been proved successful. Directions of future work, including the burgeoning use of machine learning techniques, are delineated.