A longstanding conjecture, recently solved positively by Chen, Gounelas and Liedtke, states that every projective $K3$ surface, as well as every Enriques surface, contains infinitely many rational curves. In this talk, we investigate the geometric properties of the rational curves lying on Enriques surfaces. We detect the algebraic class of many of them and in particular show that, for every natural number $k$ congruent to $1$ modulo $4$, the general Enriques surface admits irreducible rational curves of arithmetic genus $k$. We discuss the implications of our results for the geometry of rational curves on some elliptic $K3$ surfaces.