  We study ambiguities in the precise formulation of the Wheeler-DeWitt
equation for the wavefunction of the Universe that arise due to different
operator orderings in the quantum Hamiltonian. We first examine the simpler
case of the 1-dimensional minisuperspace model and derive the inner product
measure that renders the Hamiltonian hermitian. Based on this, we establish
that the Hilbert space inner products and quantum probabilities are universal,
free of any ambiguities, at the semiclassical level. Recasting the
Wheeler-DeWitt equation in a form invariant under field redefinitions of the
minisuperspace variable, we show that all ambiguity functions are contained in
a higher order scalar function, which can be used to define classes of models
with universal predictions to all orders in $\hbar$. We then generalize to
minisuperspace models of arbitrary dimension, upon the inclusion of an
arbitrary number of scalar matter fields. We show that the hermiticity of the
Hamiltonian in these cases provides a set of constraints, which can be used to
cast the WDW equation in a covariant form and establish the universality of the
inner products at the semiclassical level. In these cases as well, all
ambiguity functions appear in a higher order scalar function of the
minisuperspace manifold, which leads to distinct universality classes.
