We extend the theory of cosmological perturbations to the case when the "matter" Lagrangian is an arbitrary function of the scalar field and its first derivatives. In particular, this extension provides a unified description of known cases such as the usual scalar field and the hydrodynamical perfect fluid. In addition, it applies to the recently proposed k-inflation, which is driven by non-minimal kinetic terms in the Lagrangian. The spectrum of quantum fluctuations for slow-roll and power law k-inflation is calculated. We find, for instance, that the usual "consistency relation" between the tensor spectral index and the relative amplitude of scalar and tensor perturbations is modified. Thus, at least in principle, k-inflation is phenomenologically distinguishable from standard inflation.Introduction. The quantum theory of cosmological perturbations is one of the most interesting examples of quantum field theory in an external classical field. Indeed, the spectrum of metric perturbations generated during inflation is probably the only prediction of quantum field theory in external fields (besides the Casimir effect) which can be tested "experimentally" in the near future. The quantum theory of linearized cosmological perturbations is well developed in the case of a usual scalar field with the standard kinetic

We discuss the weak gravitational field created by isolated matter sources in the Randall-Sundrum brane world. For the case of a single wall of positive tension, the field stays localized near the wall if the source is stationary. We calculate the leading Kaluza-Klein corrections to the linearized gravitational field of a nonrelativistic spherical object, which is different from the Schwarzschild solution at large distances. In the case of two branes of opposite tension, linearized Brans-Dicke (BD) gravity is recovered on either wall, with different BD parameters. On the wall with positive tension the BD parameter is larger than 3000 provided that the separation between walls is larger than 4 times the AdS radius. The gravitational field due to shadow matter is also considered.