  Let $V$ be a vector bundle over a smooth curve $C$. In this paper, we study
twisted Brill--Noether loci parametrising stable bundles $E$ of rank $n$ and
degree $e$ with the property that $h^0 (C, V \otimes E) \ge k$. We prove that,
under conditions similar to those of Teixidor i Bigas and of Mercat, the
Brill-Noether loci are nonempty, and in many cases have a component which is
generically smooth and of the expected dimension. Along the way, we prove the
irreducibility of certain components of both twisted and "nontwisted"
Brill--Noether loci. We describe the tangent cones to the twisted Brill-Noether
loci. We end with an example of a general bundle over a general curve having
positive-dimensional twisted Brill--Noether loci with negative expected
dimension.
