Homological mirror symmetry predicts that the derived category of coherent sheaves on a curve has a symplectic counterpart as the Fukaya category of a mirror space. However, with the exception of elliptic curves, this mirror is usually a symplectic Landau-Ginzburg model, i.e. a non-compact manifold equipped with the extra data of a "stop" in its boundary at infinity. Most of the talk will focus on a family of Landau-Ginzburg models which provide mirrors to curves in (C*)^2 or in toric surfaces (or more generally to hypersurfaces in toric varieties), and their fiberwise wrapped Fukaya categories (joint work with Mohammed Abouzaid). I will then discuss more a speculative way of constructing mirrors of curves without Landau-Ginzburg models, involving a new flavor of Lagrangian Floer theory in trivalent configurations of Riemann surfaces (joint work with Alexander Efimov and Ludmil Katzarkov).