The HNC (hypernetted-chain) equation is generalized at a three-body level based on the three-body density functional theory and the Percus' idea. Supplementing with the generalized Ornstein-Zernike relations, we derive the three-body HNC theory for two- and three-body correlation functions. We solved the three-body HNC theory numerically for the case of a one-dimensional fluid and obtained the solution self-consistently. Our results are compared with the (two-body) HNC theory and molecular dynamics simulations. It is found that the three-body HNC theory improves the HNC one from the viewpoint of the radial distribution function.