A magnetic vibration absorber (MVA), completely relying on magnetic forces, is used to reduce the displacement of a vibrating structure. The distinctive feature of this absorber is the ability of tuning its linear stiffness together with its nonlinear cubic and quintic stiffnesses. Repulsive and corrective magnets are used to finely tune the values of these stiffness coefficients. A modelisation, relying on a multipolar expansion of the magnetic fields of each magnets, is used to predict the values of the stiffnesses from the geometry. Using only three geometrical parameters the MVA can be passively designed either as a nonlinear tuned vibration absorber (NLTVA), a nonlinear energy sink (NES), or a bi-stable absorber with negative linear stiffness.