A coherent system on an algebraic variety (or scheme) consists of a coherent sheaf together with a subspace of its space of sections. For coherent systems on algebraic curves, one can introduce a concept of stability depending on a real parameter and construct corresponding moduli spaces. As the parameter varies, the stability condition can change only at a discrete (and in fact finite) set of values. For rank 2 bundles with one section, the situation has been thoroughly analysed by Thaddeus and used to give a proof of the Verlinde conjectures and other useful information on the moduli spaces of bundles on curves. In this talk I will be describing joint work (still in progress) with S. Bradlow, O. Garcia-Prada and V. Munoz to analyse at least some aspects of the general case.