Lift forces are widespread in hydrodynamics. These are typically observed for big and fast objects, and are often associated with a combination of fluid inertia (\textit{i.e.} large Reynolds numbers) and some specific symmetry-breaking mechanism. In contrast, low-Reynolds-number flows are usually overdamped and do not exhibit such peculiar and interesting features. However, the inclusion of boundary effects qualitatively changes this picture. Indeed, in the context of soft and biological matter, recent studies have revealed the emergence of novel lift forces generated by boundary softness, flow gradients and/or surface charges. The aim of the present review is to gather and analyse this corpus of literature, in order to identify and unify the questioning within the associated communities, and pave the way towards future research around lift effects at low Reynolds numbers.