Applications of ionic liquids at electrified interfaces to energy-storage systems, electrowetting devices, or nanojunction gating media cannot proceed without a deep understanding of the structure and properties of the interfacial double layer. This article provides a detailed critique of the present work on this problem. It promotes the point of view that future considerations of ionic liquids should be based on the modern statistical mechanics of dense Coulomb systems, or density-functional theory, rather than classical electrochemical theories which hinge on a dilute-solution approximation. The article will, however, contain more questions than answers. To trigger the discussion, it starts with a simplified original result. A new analytical formula is derived to rationalize the potential dependence of double-layer capacitance at a planar metal-ionic liquid interface. The theory behind it has a mean-field character, based on the Poisson-Boltzmann lattice-gas model, with a modification to account for the finite volume occupied by ions. When the volume of liquid excluded by the ions is taken to be zero (that is, if ions are extremely sparsely packed in the liquid), the expression reduces to the nonlinear Gouy-Chapman law, the canonical result typically used to describe the potential dependence of capacitance in electrochemical double layers. If ionic volume exclusion takes more realistic values, the formula shows that capacitance-potential curves for an ionic liquid may differ dramatically from the Gouy-Chapman law. Capacitance has a maximum close to the potential of zero charge, rather than the familiar minimum. At large potenials, capacitance decreases with the square root of potential, rather than increases exponentially. The reported formula does not take into account the specific adsorption of ions, which, if present, can complicate the analysis of experimental data. Since electrochemists use to think about the capacitance data in terms of the classical Gouy-Chapman theory, which, as we know, should be good only for electrolytes of moderate concentration, the question of which result is "better" arises. Experimental data are sparse, but a quick look at them suggests that the new formula seems to be closer to reality. Opinions here could, however, split. Indeed, a comparison with Monte Carlo simulations has shown that incorporation of restricted-volume effects in the mean-field theory of electrolyte solutions may give results that are worse than the simple Gouy-Chapman theory. Generally, should the simple mean-field theory work for such highly concentrated ionic systems, where the so-called ion-correlation effects must be strong? It may not, as it does not incorporate a possibility of charge-density oscillations. Somehow, to answer this question definitely, one should do further work. This could be based on density-functional theory (and possibly not on what is referred to as local density approximation but rather "weighted density approximation"), field theory methods for the account of fluctuations in the calculation of partition function, heuristic integral equation theory extended to the nonlinear response, systematic force-field computer simulations, and, most importantly, experiments with independently determined potentials of zero charge, as discussed in the paper.