38 pages. Nr. 5 is final version (02/08/2006), one typo was corrected and one reference deleted. Version nr. 4 (12/06/2006) included minor corrections and two added references. Version nr. 3 (23/06/2005) had appendix and new references added.Let C be an algebraic curve of genus g ≥ 2. A coherent system on C consists of a pair (E,V), where E is an algebraic vector bundle over C of rank n and degree d and V is a subspace of dimension k of the space of sections of E. The stability of the coherent system depends on a parameter α. We study the geometry of the moduli space of coherent systems for different values of α when k ≤ n and the variation of the moduli spaces when we vary α. As a consequence, for sufficiently large α, we compute the Picard groups and the first and second homotopy groups of the moduli spaces of coherent systems in almost all cases, describe the moduli space for the case k = n − 1 explicitly, and give the Poincaré polynomials for the case k = n − 2. In an appendix, we describe the geometry of the “flips” which take place at critical values of α in the simplest case, and include a proof of the existence of universal families of coherent systems when GCD(n, d, k) = 1.All authors are members of the research group VBAC (Vector Bundles on Algebraic Curves), which was supported by EAGER (EC FP5 Contract no. HPRN-CT-2000-00099) and by EDGE (EC FP5 Contract no. HPRN-CT-2000-00101). Support was also received from two grants from the European Scientific Exchange Programme of the Royal Society of London and the Consejo Superior de Investigaciones Científicas (15455 and 15646). The first author was partially supported by the National Science Foundation under grant DMS-0072073.Peer reviewed