Let $C$ be a smooth irreducible projective curve. We can consider the Brill-Noether locus $B(2,\omega_C,r)$ of stable rank two vector bundles with canonical determinant and at least $r$ linearly independent sections on $C$. There is a complete description of $B(2,\omega_C,r)$ when $C$ has general moduli and genus $g\leq 12$. For higher genus several basic questions like non-emptyness, irreducibility, etc, are still open. The talk will focus in these questions and some conjectures.