A natural question, arising from physics discussions of two-dimensional quantum gravity, is whether there exists a good notion of a random flat metric on the unit disk. In this talk I will present the combinatorial family of rigid quadrangulations, that is expected to feature such a universal random metric in a scaling limit. The enumeration or rigid quadrangulations is solved via a bijection with certain integer-labeled quadrangulations featuring in works of Bousquet-Mélou and Elvey Price. We will hint on implications of this bijection for the limit. Based on arXiv:2509.24785.