The quest for materials possessing both a magnetic ordering temperature above room temperature and a large electrical polarization is an important research direction in order to design novel spintronic and memory devices. Up to now, BiFeO3 and related systems are the only known compounds simultaneously possessing such characteristics. Here, first-principles calculations predict that another family of materials, namely epitaxial films made of rare-earth orthoferrites (RFeO3), can also exhibit such desired features. As a matter of fact, applying a large enough strain to these compounds, which are nominally paraelectric and have a high magnetic transition temperature, is predicted to render them ferroelectric, and thus multiferroic. At high compressive strain, the resulting ferroelectric phase of RFeO3 systems having large rare-earth ions is even a tetragonal state characterized by a giant polarization and axial ratio. For large tensile strain, two striking inhomogenous ferroelectric phases--including one never observed before in any perovskite--are further predicted as having significant polarization. A multiphase boundary also occurs, which may lead to optimization of properties or unusual features. Finally, many quantities, including electrical polarization and magnetic ordering temperature, are tunable by varying the epitaxial strain and/or chemical pressure.