  The construction of physical models with local time-reparametrization
invariance is reviewed. Negative-energy contributions to the hamiltonian are
shown to be crucial for the realization of this reparametrization symmetry. The
covariant formulation of the dynamics is used to develop a time and gauge
invariant Hamilton-Jacobi theory. This formalism is applied to solve for the
cosmology of a homogeneous universe of the Friedmann-Lemaitre-Robertson-Walker
type. After a discussion of empty universes, the FLRW theory is extended with
homogeneous scalar fields generically described by a $\sg$-model on some scalar
manifold. An explicit gauge-invariant solution is constructed for the
non-linear O(N)-models.
