Let SU C (r) be the moduli space of vector bundles of rank r and trivial determinant on a curve C . A general E in SU C (r) defines a divisor Θ E in the linear system | r Θ|, where Θ is the canonical theta divisor in Pic g -1 ( C ). This defines a rational map Θ: SU C (r) → | r Θ|, which turns out to be the map associated to the determinant bundle on SU C (r) (the positive generator of Pic ( SU C (r) ). In genus 2 we prove that this map is generically finite and dominant. The same method, together with some classical work of Morin, shows that in rank 3 and genus 3 the theta map is a finite morphism - in other words, every vector bundle in SU C (3) admits a theta divisor.