  We derive, in 3+1 spacetime dimensions, two alternative systems of
quasi-linear wave equations, based on Friedrich's conformal field equations. We
analyse their equivalence to Einstein's vacuum field equations when appropriate
constraint equations are satisfied by the initial data. As an application, the
characteristic initial value problem for the Einstein equations with data on
past null infinity is reduced to a characteristic initial value problem for
wave equations with data on an ordinary light-cone.
