Let $\Sigma$ be a simply connected surface properly embedded in $\mathbb{H}^2 \times \mathbb{R}$ with constant mean curvature $1/2$. In this talk, we show that, under additional hypotheses on its angle function or on its asymptotic boundary, $\Sigma$ is either an entire graph over $\mathbb{H}^2$ or a horocylinder. Joint work with L. Hauswirth.