An asymptotic theory of surface waves trapped on vertically uniform jet currents is developed as a first step towards a systematic description of wave dynamics on oceanic jet currents. It has been shown that in a linear setting an asymptotic separation of vertical and horizontal variables, which underpins the modal description of the wave field on currents, is possible if either the current velocity is small compared to the wave celerity or the current width is large compared to the wavelength along the current. The scheme developed enables us to obtain solutions as an asymptotic series with any desired accuracy. The initially three-dimensional problem is reduced to solving one-dimensional equations with the lateral and vertical dependence being prescribed by the corresponding modal structure. To leading order in current magnitude to wave celerity, the boundary value problem specifying the modes and eigenvalues reduces to classical Sturm–Liouville type based upon the one-dimensional stationary Schrödinger equation. The modes, both trapped and ‘passing-through’, form a complete orthogonal set. This makes the modal description of waves on currents a mathematically attractive alternative to the approaches currently adopted. Properties of trapped eigenmodes and their dispersion relations are examined both for broad currents of arbitrary magnitude, where the modes are not orthogonal, and for weak currents, where the modes are orthogonal. Several model profiles for which nice analytical solutions of the leading-order boundary value problem are known were used to get an insight. The asymptotic solutions proved not only to capture qualitative behaviour well but also to provide a good quantitative description even for unrealistically strong and narrow currents. The results are discussed for various oceanic currents, with particular attention paid to the Agulhas Current, for which specific estimates were derived. For typical dominant wind waves and swell, all oceanic-jet-type currents are weak and, correspondingly, the developed asymptotic scheme based upon one-dimensional stationary Schrödinger equation for modes applies.