In recent microfluidic experiments with solutions of bacteria we observed the formation of biofilms in the form of thread-like structures, called ‘streamers’, which float in the middle plane of the channel and are connected to the side walls at the inner corners. Motivated by this observation, we discuss here the pressure-driven low-Reynolds-number flow around a corner bounded by the walls of a channel with rectangular cross-section. We numerically solve the flow field in a channel of constant cross-section, which exhibits 90° sharp corners, or turns with constant curvature, or portions with slowly changing curvature along the flow direction, for finite, but small, values of the Reynolds numbers and including the limit of vanishingly small Reynolds numbers. In addition, we develop a matched asymptotic expansion solution for the flow around two boundaries intersecting at an angle α and spanning the small gap h between two horizontal plates. We illustrate the basic features of the flow in these channel geometries by describing the three-dimensional velocity field and the distribution of streamwise vorticity and helicity, and comparing the numerical solutions with predictions based on the asymptotic approach. We demonstrate that near a corner or a change in the curvature of the side wall the flow is three-dimensional and pairs of counter-rotating vortical structures are present, as identified by Balsa (J. Fluid Mech., vol. 372, 1998, p. 25). Finally, we discuss how this secondary flow depends on the significant geometric parameters, the aspect ratio of the channel cross-section, the radius of curvature of the turn and, more generally, the variation of the curvature of the channel side boundary. We believe that these three-dimensional secondary flow structures are relevant to transport problems where accumulation of material at the boundary is possible.