The quantum anomalous Hall effect (QAHE) is a fundamental transport phenomenon in the field of condensed-matter physics. Without an external magnetic field, spontaneous magnetization combined with spin-orbit coupling gives rise to a quantized Hall conductivity. So far, a number of theoretical proposals have been made to realize the QAHE, but all based on inorganic materials. Here, using first-principles calculations, we predict a family of 2D organic topological insulators for realizing the QAHE. Designed by assembling molecular building blocks of triphenyl-transition-metal compounds into a hexagonal lattice, this new class of organic materials is shown to have a nonzero Chern number and exhibits a gapless chiral edge state within the Dirac gap.