Abstract. 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.

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