  We propose a novel codimension-n holography, called cone holography, between
a gravitational theory in $(d+1)$-dimensional conical spacetime and a 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/R\'enyi
entropy and correlation functions, and find good agreements 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/R\'enyi entropy and
AdS/BCFT(dCFT). Thus it is expected to have a wide range of applications.
