The moduli of curves with W-structure and their corresponding cohomological field theories form an orbifolded Landau-Ginzburg A-model and are the subject of several beautiful mirror symmetry conjectures. Some of these conjectures have been proved, including the Witten ADE-integrable hierarchies conjecture, while others are still open. In this talk I will give an overview of the theory as well as discussing recent progress on some of the conjectures.