The moduli space of stable curves parameterizes tuples $(C,p_1,...,p_n)$ of a compact, complex curve $C$ together with distinct marked points $p_1,\dots, p_n$. Inside this moduli space, there are natural subsets, called the strata of $k$-differentials, defined by the condition that there exists a meromorphic $k$-differential on $C$ with zeros and poles of some fixed multiplicities at the points $p_i$. I will discuss basic properties of these strata and explain a conjecture relating their fundamental class to the so-called double ramification cycles on the moduli space. I explain the idea of the proof of this conjecture and some ongoing work with Costantini and Sauvaget on how to use this relation to compute intersection numbers of the strata with $\psi$-classes on the moduli of curves.