Recent studies have demonstrated the potential of antiferromagnets as the active component in spintronic devices. This is in contrast to their current passive role as pinning layers in hard disk read heads and magnetic memories. Here we report the epitaxial growth of a new high-temperature antiferromagnetic material, tetragonal CuMnAs, which exhibits excellent crystal quality, chemical order and compatibility with existing semiconductor technologies. We demonstrate its growth on the III-V semiconductors GaAs and GaP, and show that the structure is also lattice matched to Si. Neutron diffraction shows collinear antiferromagnetic order with a high Néel temperature. Combined with our demonstration of room-temperature-exchange coupling in a CuMnAs/Fe bilayer, we conclude that tetragonal CuMnAs films are suitable candidate materials for antiferromagnetic spintronics.