In any quantum theory, in which the metric tensor of Einstein’s gravitational theory is also quantized, it becomes meaningless to ask for an initial space-like surface on which to specify the conventional field commutators. The covariant quantum formalism, in which all fields either commute or fail to do so only when the field’s points coincide, is proposed as being suitable to quantize gravity. The extension of the covariant quantum formalism to general boson fields that interact in an intrisically nonlinear way with external fields is analysed in some detail. This formalism is applied to the case of the free gravitational field. In a functional representation, the measure on metrics is found to be that proposed by Misner. A basic state of the quantized gravitational theory is proposed, which involves a summation over all permissible metrics in the entire space-time manifold.