Hyperkähler varieties form one of the three building blocks for projective varieties with trivial canonical bundle. Their classification is widely open, but there is growing evidence that the problem is closely related to the classification of 2- dimensional Calabi-Yau categories, with the most optimistic hope being that any hyperkähler variety arises from a suitable moduli space of objects in such a category. I will discuss this circle of ideas, including work in progress with Arend Bayer, Laura Pertusi, and Xiaolei Zhao completing this program for hyperkähler varieties of Kummer type.