In this talk, I will discuss recent advances towards overcoming the sign problem of Monte Carlo simulations. There are several parameter regimes of lattice quantum field theory that cannot be simulated with conventional Monte Carlo methods, including topological terms, non-zero baryon density, and real-time dynamics. At the example of (1+1)-dimensional benchmark models, I will discuss new directions towards overcoming these problems, based on tensor networks and quantum computing. Finally, I will comment on the requirements to integrate these methods into (3+1)-dimensional lattice simulations in the future, focusing on Lattice QCD.