A short review on the compositeness of the $X(3872)$

The $X(3872)$ could be a shallow $D\bar D^*$ bound state, a compact four-quark state, or a partially composite particle, i.e. a superposition of the two. We will review how these hypotheses could be tested experimentally, examining especially the cases in which the $X$ is a pure bound state or a pure compact tetraquark. Data on $X\to D\bar D\pi$ decays are compared with the analysis of the $X$ lineshape. The pure bound state hypothesis corresponds to a well-defined region in parameter space defined by the width of the $D^*$ versus the binding energy of the $X$. As for the $X$ lineshape, we observe that the currently available experimental analysis tests the compatibility with the compact hypothesis for the $X$. We propose how to extend the analysis to examine the molecular or the partially composite hypotheses. We also review the analysis on the radiative decays of the $X$ including pion corrections confirming some conclusions reached in the literature on the use of the universal wave function description for the molecular $X$.