Given a Riemann surface, we can equip it with a k-differential which can be written in any local holomorphic
coordinate z on X as f(z)dz^k where f is meromorphic. When the singularities are all of orders > -k, a k-differential corresponds to a collection of polygons in the complex plane with sides identified pairwise by translations and rotations by 2pi/k. When k=1, these objects are famously called translation surfaces. More often than not, k-differentials are studied within a stratification by the number of singularities and their orders. Strata of k-differentials, especially when k=1,2, has been the focus of many for decades and come with a rich variety of questions which have been asked, such as their topology and intrinsic and extrinsic dynamical properties of their surfaces. However, very little is known about strata of k-differentials when k>2… Until now! In this talk, we discuss the major breakthrough of a classification of components of strata of k-differentials. If time permits, we will discuss orbit closures and counting functions on generic surfaces in a stratum, answering the positive genus analogue of a long term goal of Mirzakhani-Wright. The key to unlocking these results is the existence of a Euclidean cylinder in each component of a stratum of positive genus surfaces. This is joint work with Paul Apisa.