Predictions for cosmological observables, such as the scalar and tensor power spectra, their spectral indices and their higher order runnings, suffer from the so-called frame problem. They depend on whether the computation has been performed in the Jordan or Einstein frame. In this talk, I will present a frame-covariant formalism which enables one to address the frame problem in the context of general scalar-curvature multifield theories. The formalism makes use of notions and techniques known from differential geometry, and can be extended beyond the tree approximation in the Vilkovisky-De Witt framework which needs to be generalized appropriately. I will show how frame-invariant predictions for cosmological observables can be obtained in simple two-field models, motivated by Higgs inflation.