We propose a novel codimension- n holography, called cone holography, between a gravitational theory in (d + 1)-dimensional conical spacetime and a conformal field theory (CFT) on the (d + 1 − n) -dimensional defects. Similar to wedge holography, the cone holography can be obtained by taking the zero-volume limit of holographic defect CFT. Remarkably, it can be regarded as a holographic dual of the edge modes on the defects. For one class of solutions, we prove that the cone holography is equivalent to AdS / CFT , by showing that the classical gravitational action and thus the CFT partition function in large N limit are the same for the two theories. In general, cone holography and AdS / CFT are different due to the infinite towers of massive Kaluza-Klein modes on the branes. We test cone holography by studying Weyl anomaly, entanglement/Rényi entropy, and correlation functions and find good agreement between the holographic and the CFT results. In particular, the c -theorem is obeyed by cone holography. These are strong supports for our proposal. We discuss two kinds of boundary conditions, the mixed boundary condition and Neumann boundary condition, and find that they both define a consistent theory of cone holography. We also analyze the mass spectrum on the brane and find that the larger the tension is, the more continuous the mass spectrum is. The cone holography can be regarded as a generalization of the wedge holography, and it is closely related to the defect CFT, entanglement/Rényi entropy, and AdS/BCFT(dCFT). Here BCFT(dCFT) means a conformal field theory defined in a manifold with a boundary (defect). Thus, it is expected to have a wide range of applications.