Abstract: Let V be a vector bundle over a smooth curve C. The twisted Brill-Noether locus B^k_{n, e} (V) parametrises stable bundles of rank n and degree e such that V \otimes E has at least k independent sections. This is a determinantal variety, whose tangent spaces are controlled by a Petri trace map. Generalising a construction of Mercat, we show for many values of the parameters that B^k_{n, e} (V) is nonempty. When C and V are general, we show that under certain numerical conditions, B^k_{n, e} (V) has a component which is generically smooth and of the expected dimension. (Joint work with Michael Hoff (Saarbrücken) and Peter Newstead (Liverpool))