A Lagrange-multiplier based approach is presented for the general solution of multi-body contact within an explicit ÿnite element framework. The technique employs an explicit predictor step to permit the detection of interpenetration and then utilizes a corrector step, whose solution is obtained with a pre-conditioned matrixfree conjugate gradient projection method, to determine the Lagrange multipliers necessary to eliminate the predicted penetration. The predictor-corrector algorithm is developed for deformable bodies based upon the central di erence method, and for rigid bodies from momentum and energy conserving approaches. Both frictionless and Coulomb-based frictional contact idealizations are addressed. The technique imposes no timestep constraints and quickly mitigates velocity discontinuities across closed interfaces. Special attention is directed toward contact between rigid bodies. Algorithmic moment arms conserve the translational and angular momentums of the system in the absence of external loads. Elastic collisions are captured with a two-phase predictor-corrector approach and a geometrically approximate velocity jump criterion. The ÿrst step solves the inelastic contact problem and identiÿes inactive constraints between rigid bodies, while the second step generates the necessary velocity jump condition on the active constraints. The velocity criterion is shown to algorithmically preserve the system kinetic energy for two unconstrained rigid bodies.