The authors propose a new covariant and gauge-invariant (GI) treatment of perturbations in a Robertson-Walker universe dominated by a classical scalar field phi . They first set up the formalism, based on the natural slicing of the problem by the surfaces phi =constant, and introduce a set of covariantly defined GI variables. In their approach the whole inhomogeneity of the matter field is incorporated in the GI spatial fluctuations of the momentum psi of phi ; then the GI density perturbations are simply proportional to the momentum perturbations. The inhomogeneity of the geometry is characterized by GI fluctuations of the 3-curvature scalar of the surfaces phi =constant. The time evolution of the matter and curvature perturbations are coupled by a pair of first-order linear differential equations. Correspondingly, each GI variable satisfies a second-order linear homogeneous differential equation. When the background curvature vanishes, k=0, the curvature variable is conserved for perturbation scales larger than the horizon, but this is no longer true in general if k not=0. They discuss simple examples, including the case when more than one scalar field is present, recovering standard results for inflationary universe models. They also demonstrate that in coasting solutions with k=-1, inhomogeneities are damped out on all scales.