The effect of epitaxial compressive and tensile strains on the physical properties of multiferroic films is investigated via the use of ab initio calculations. Striking phenomena are predicted. Examples include (1) the existence of a previously overlooked strain-induced low-symmetry phase bridging the tetragonal (compressively strained) and orthorhombic (tensile-strained) ferroelectric structures, which exhibits a very large piezoelectric response; (2) a first-order magnetic transition within the tetragonal polar state, which is accompanied by remarkable changes in polarization and out-of-plane lattice constant; (3) the existence of a state that is both ferromagnetic and ferroelectric with a large polarization and axial ratio at large enough compressive strain; and (4) the possibility to induce a structural phase transition between states of different symmetries (e.g., between tetragonal and monoclinic, or between orthorhombic and monoclinic) by applying a magnetic field, which constitutes an original magnetoelectric effect. Experiments are called for to confirm such predictions.