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AMIGA is a publicly available adaptive mesh refinement code for (dissipationless) cosmological simulations. It combines an N-body code with an Eulerian grid-based solver for the full set of magnetohydrodynamics (MHD) equations in order to conduct simulations of dark matter, baryons and magnetic fields in a self-consistent way in a fully cosmological setting. Our numerical scheme includes effective methods to ensure proper capturing of shocks and highly supersonic flows and a divergence-free magnetic field. The high accuracy of the code is demonstrated by a number of numerical tests.