For a negatively curved compact Riemannian manifold (or more generally, for an Anosov flow), the Ruelle zeta function is defined by $$\zeta(s)=\prod_\gamma (1-e^{-s\ell_\gamma} ),\quad \Re s\gg 1,$$ where the product is taken over all primitive closed geodesics $\gamma$ with $\ell_\gamma>0$ denoting their length. Remarkably, this zeta function continues meromorphically to all of $ \mathbb C$. Using recent advances in the study of resonances for Anosov flows and simple arguments from microlocal analysis, we prove that for an orientable negatively curved surface, the order of vanishing of $\zeta(s)$ at $s=0$ is given by the absolute value of the Euler characteristic. In constant curvature this follows from the Selberg trace formula and this is the first result of this kind for manifolds which are not locally symmetric. This talk is based on joint work with Maciej Zworski.