  We carry out the programme of R. Liouville \cite{Liouville} to construct an
explicit local obstruction to the existence of a Levi--Civita connection within
a given projective structure $[\Gamma]$ on a surface. The obstruction is of
order 5 in the components of a connection in a projective class. It can be
expressed as a point invariant for a second order ODE whose integral curves are
the geodesics of $[\Gamma]$ or as a weighted scalar projective invariant of the
projective class. If the obstruction vanishes we find the sufficient conditions
for the existence of a metric in the real analytic case. In the generic case
they are expressed by the vanishing of two invariants of order 6 in the
connection. In degenerate cases the sufficient obstruction is of order at most
8.
