Advection–diffusion equations occur in a wide variety of fields in many contexts of active and passive transport in fluids and plasmas. The effects of sheared advective flows in the presence of irreversible processes such as diffusion and viscosity are of considerable current interest in tokamak and astrophysical contexts, where they are thought to play a key role in both transport and the dynamical structures characteristic of electromagnetic plasma turbulence. In this paper we investigate the spectral and evolutionary properties of relatively simple, linear, advection–diffusion equations. We apply analytical approaches based on standard Green function methods to obtain insight into the nature of the spectra when the advective and diffusive effects occur separately and in combination. In particular, the physically interesting limit of small (but finite) diffusion is studied in detail. The analytical work is extended and supplemented by numerical techniques involving a direct solution of the eigenvalue problem as well as evolutionary studies of the initial-value problem using a parallel code, CADENCE. The three approaches are complementary and entirely consistent with each other when an appropriate comparison is made. They reveal different aspects of the properties of the advection–diffusion equation, such as the ability of sheared flows to generate a direct cascade to high wave numbers transverse to the advection and the consequent enhancement of even small amounts of diffusivity. The invariance properties of the spectra in the low diffusivity limit and the ability of highly sheared, jet-like flows to ‘confine’ transport to low shear regions are demonstrated. The implications of these properties in a wider context are discussed and set in perspective.