This paper extends the theory of cosmological perturbations to the case when the "matter" Lagrangian is an arbitrary function of the scalar field and its first derivatives. This extension provides a unified description of known cases such as the usual scalar field and the hydrodynamical perfect fluid. It also 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. The paper finds that the usual "consistency relation" between the tensor spectral index and the relative amplitude of scalar and tensor perturbations is modified. Thus, k-inflation is phenomenologically distinguishable from standard inflation. The paper considers the most general local action for a scalar field coupled to Einstein gravity, which involves at most first derivatives of the field. The Lagrangian for the scalar field is called p because it plays the role of pressure. The energy momentum tensor is derived from the matter Lagrangian. The paper then considers the background of an expanding Friedmann Universe with arbitrary spatial curvature. Two independent equations for two unknown background variables are written down in the familiar form. The paper then considers the equations for perturbations, considering small inhomogeneities of the scalar field and taking into account that the perturbations in the components of the energy momentum tensor can be expressed in terms of the perturbation of the scalar field and the gravitational potential. The linearized equations for the perturbations are derived and simplified in two important cases: a) in spatially flat Universe and b) for hydrodynamical matter or scalar field without potential. The paper then considers the action for perturbations, which is needed to normalize the amplitude of quantum fluctuations. The action is obtained by expanding the action to second order in perturbations. The action is then compared with previously derived actions for hydrodynamical matter and scalar field with minimal kinetic term in K=0 and K≠0 universes. The first order action which reproduces the equations of motion is derived. The paper then considers the power spectra for the quantity ζ = v/z, which is directly related to the gravitational potential. The scalar spectral index is calculated and found to be different from the usual result in standard inflation. The tensor spectral index is also calculated and found to be different from the usual result in standard inflation. The ratio of tensor to scalar amplitudes is also calculated and found to be different from the usual result in standard inflation. The paper then considers power law k-inflation, which is driven by a Lagrangian of the form p = g(X)φ⁻². The solutions are characterized by X = X₀ = const, where X₀ is a solution of a certain equation. The power exponent is given by β = 4πGg,X(X₀). The speed of sound is a constant and can be adjusted to any given value. The paperThis paper extends the theory of cosmological perturbations to the case when the "matter" Lagrangian is an arbitrary function of the scalar field and its first derivatives. This extension provides a unified description of known cases such as the usual scalar field and the hydrodynamical perfect fluid. It also 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. The paper finds that the usual "consistency relation" between the tensor spectral index and the relative amplitude of scalar and tensor perturbations is modified. Thus, k-inflation is phenomenologically distinguishable from standard inflation. The paper considers the most general local action for a scalar field coupled to Einstein gravity, which involves at most first derivatives of the field. The Lagrangian for the scalar field is called p because it plays the role of pressure. The energy momentum tensor is derived from the matter Lagrangian. The paper then considers the background of an expanding Friedmann Universe with arbitrary spatial curvature. Two independent equations for two unknown background variables are written down in the familiar form. The paper then considers the equations for perturbations, considering small inhomogeneities of the scalar field and taking into account that the perturbations in the components of the energy momentum tensor can be expressed in terms of the perturbation of the scalar field and the gravitational potential. The linearized equations for the perturbations are derived and simplified in two important cases: a) in spatially flat Universe and b) for hydrodynamical matter or scalar field without potential. The paper then considers the action for perturbations, which is needed to normalize the amplitude of quantum fluctuations. The action is obtained by expanding the action to second order in perturbations. The action is then compared with previously derived actions for hydrodynamical matter and scalar field with minimal kinetic term in K=0 and K≠0 universes. The first order action which reproduces the equations of motion is derived. The paper then considers the power spectra for the quantity ζ = v/z, which is directly related to the gravitational potential. The scalar spectral index is calculated and found to be different from the usual result in standard inflation. The tensor spectral index is also calculated and found to be different from the usual result in standard inflation. The ratio of tensor to scalar amplitudes is also calculated and found to be different from the usual result in standard inflation. The paper then considers power law k-inflation, which is driven by a Lagrangian of the form p = g(X)φ⁻². The solutions are characterized by X = X₀ = const, where X₀ is a solution of a certain equation. The power exponent is given by β = 4πGg,X(X₀). The speed of sound is a constant and can be adjusted to any given value. The paper