Implementation of multi-particle unitarity is among the biggest challenges in the nonperturbative S-matrix bootstrap. Unitarity implies the existence of branch cuts along normal thresholds. Once combined with analyticity, unitarity also predicts the existence of other non-trivial singularities known as Landau curves. In the elastic domain, all the Landau curves are known thanks to Mandelstam. In this talk, I will show how the leading multi-particle Landau curves can be obtained using graph-theoretic techniques. To our surprise, we find an infinite sequence of Landau curves that accumulate at finite energies. Our results should apply to any quantum field theory with at least one massive particle.