  We define non-pluripolar products of closed positive currents on a compact
Kaehler manifold. We show that a positive non-pluripolar measure can be written
in a unique way as the top degree self-intersection (in the non-pluripolar
sense) of a closed positive current in given big cohomology class. The solution
is shown to have minimal singularities in the sense of Demailly if the measure
is regular enough. These results are combined with a fixed point argument to
construct singular Kaehler-Einstein volume forms with minimal singularities on
varieties of general type.
