Using first-principles calculations within density functional theory, we investigate the band topology of ternary chalcopyrites of composition I-III-VI2 and II-IV-V2. By exploiting adiabatic continuity of their band structures to the binary 3D-HgTe, combined with direct evaluation of the Z2 topological invariant, we show that a large number of chalcopyrites can realize the topological insulating phase in their native states. The ability to host room-temperature ferromagnetism in the same chalcopyrite family makes them appealing candidates for novel spintronics devices.