  Let $X$ be a smooth irreducible projective curve of genus $g$ and gonality 4.
We show that the canonical model of $X$ is contained in a uniquely defined
surface, ruled by conics, whose geometry is deeply related to that of $X$. This
surface allows us to define four invariants of $X$ and hence to stratify the
moduli space of four-gonal curves by means of closed irreducible subvarieties
whose dimensions we compute.
