We consider the recently introduced "galileon" field in a dynamical spacetime. When the galileon is assumed to be minimally coupled to the metric, we underline that both field equations of the galileon and the metric involve up to third-order derivatives. We show that a unique nonminimal coupling of the galileon to curvature eliminates all higher derivatives in all field equations, hence yielding second-order equations, without any extra propagating degree of freedom. The resulting theory breaks the generalized "Galilean" invariance of the original model.Comment: 10 pages, no figure, RevTeX4 format; v2 adds footnote 1, Ref. [12], reformats the link in Ref. [14], and corrects very minor typo

We consider "brane-universes", where matter is confined to four-dimensional hypersurfaces (three-branes) whereas one extra compact dimension is felt by gravity only. We show that the cosmology of such branes is definitely different from standard cosmology and identify the reasons behind this difference. We give a new class of exact solutions with a constant five-dimensional radius and cosmologically evolving brane. We discuss various consequences.