  In this paper, we study a class of two-dimensional Finsler metrics defined by
a Riemannian metric $\alpha$ and a 1-form $\beta$. We characterize those
metrics which are Douglasian or locally projectively flat by some equations. In
particular, it shows that the known fact that $\beta$ is always closed for
those metrics in higher dimensions is no longer true in two dimensional case.
Further, we determine the local structures of two-dimensional
$(\alpha,\beta)$-metrics which are Douglassian, and some families of examples
are given for projectively flat classes with $\beta$ being not closed.
