Abstract Usually during solidification of alloys solute is redistributed at the solid/liquid interface. For high solidification velocities a deviation from local thermodynamic equilibrium at the interface reduces the tendency for redistribution. This effect is known as solute trapping. Based on the model for binary systems presented in M.J. Aziz, T. Kaplan [Acta Metall. 36 (1988) 2335], we have developed a theory for multi-componental solute trapping. It enables the determination of the n solid concentrations by solving an n-dimensional nonlinear system of equations, called first response-functions. In addition we derived a relation between the growth velocity and the driving force for crystallisation for both planar and non-planar solid/liquid interfaces, called second response-function. Our model shows that the usual concept to calculate the interface temperature by reducing the actual “interface liquidus temperature” by a curvature and a kinetic undercooling term cannot be applied with non-dilute binary and arbitrary multi-componental alloys.