A stabilized finite element method for the simulation of instationary and stationary multi-ion transport in dilute electrolyte solutions is presented. The proposed computational approach accounts for all three ion-transport phenomena, that is, convection, diffusion and migration, as well as nonlinear electrode kinetics boundary conditions. The governing equations form a set of coupled nonlinear partial differential equations subject to an electroneutrality condition. The latter establishes an algebraic constraint to the problem formulation. Derived from the variational multiscale method, we introduce stabilization terms which prevent potential spurious oscillations arising in the convectiondominated case when a standard Galerkin finite element method is used. For various numerical examples, it is demonstrated that the proposed computational method is robust and provides accurate results.