  Let $C$ be a curve of genus $g$. A fundamental problem in the theory of
algebraic curves is to understand maps $C \to \mathbb{P}^r$ of specified degree
$d$. When $C$ is general, the moduli space of such maps is well-understood by
the main theorems of Brill--Noether theory. Despite much study over the past
three decades, a similarly complete picture has proved elusive for curves of
fixed gonality. Here we complete such a picture, by proving analogs of all of
the main theorems of Brill--Noether theory in this setting. As a corollary, we
prove a conjecture of Eisenbud and Schreyer regarding versal deformation spaces
of vector bundles on $\mathbb{P}^1$.
