  Employing alternative spacetime volume-forms (generally-covariant integration
measure densities) independent of the pertinent Riemannian spacetime metric
have profound impact in general relativity. Although formally appearing as
"pure-gauge" dynamical degrees of freedom they trigger a number of remarkable
physically important phenomena such as: (i) new mechanism of dynamical
generation of cosmological constant; (ii) new type of "quintessential
inflation" scenario in cosmology; (iii) non-singular initial "emergent
universe" phase of cosmological evolution preceding the inflationary phase;
(iv) new mechanism of dynamical spontaneous breakdown of supersymmetry in
supergravity; (v) gravitational electrovacuum "bags". We study in some detail
the properties, together with their canonical Hamiltonian formulation, of a
class of generalized gravity-matter models built with two independent
non-Riemannian volume-forms and discuss their implications in cosmology.
