We present BAM: a novel Bias Assignment Method envisaged to generate mock catalogues. Combining the statistics of dark matter tracers from a high-resolution cosmological N-body simulation and the dark matter density field calculated from down-sampled initial conditions using efficient structure formation solvers, we extract the halo-bias relation on a mesh of a 3 h^{-1} Mpc cell side resolution as a function of properties of the dark matter density field (e.g. local density, cosmic web type), automatically including stochastic, deterministic, local and non-local components. We use this information to sample the halo density field, accounting for ignored dependencies through an iterative process. By construction, our approach reaches {˜ } 1 {per cent} accuracy in the majority of the k-range up to the Nyquist frequency without systematic deviations for power spectra (about k ˜ 1 h Mpc-1) using either particle mesh or Lagrangian perturbation theory based solvers. When using phase-space mapping to compensate the low resolution of the approximate gravity solvers, our method reproduces the bispectra of the reference within 10 {per cent} precision studying configurations tracing the quasi-non-linear regime. BAM has the potential to become a standard technique to produce mock halo and galaxy catalogues for future galaxy surveys and cosmological studies being highly accurate, efficient and parameter free.