Let $C$ be an algebraic curve of genus $g\ge2$. 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 $\alpha$. We study the geometry of the moduli space of coherent systems for $0<d\le2n$. We show that these spaces are irreducible whenever they are non-empty and obtain necessary and sufficient conditions for non-emptiness.Support was received from a grant from the European Scientific Exchange Programme of the Royal Society of London and the Consejo Superior de Investigaciones Científicas (15646) and a further grant from the Royal Society of London for an International Joint Project (2005/R3). The first author was partially supported by the National Science Foundation under grant DMS-0072073. The first, second and fourth authors were supported through MEC grant MTM2004-07090-C03-01 (Spain) and the fourth also through NSF grant DMS-0111298 (US). The fifth author was supported by the Academia Mexicana de Ciencias during a visit to CIMAT, Guanajuato, Mexico.Peer reviewed