Strain engineering is a powerful tool with which to enhance semiconductor device performance1,2. Halide perovskites have shown great promise in device applications owing to their remarkable electronic and optoelectronic properties3-5. Although applying strain to halide perovskites has been frequently attempted, including using hydrostatic pressurization6-8, electrostriction9, annealing10-12, van der Waals force13, thermal expansion mismatch14, and heat-induced substrate phase transition15, the controllable and device-compatible strain engineering of halide perovskites by chemical epitaxy remains a challenge, owing to the absence of suitable lattice-mismatched epitaxial substrates. Here we report the strained epitaxial growth of halide perovskite single-crystal thin films on lattice-mismatched halide perovskite substrates. We investigated strain engineering of α-formamidinium lead iodide (α-FAPbI3) using both experimental techniques and theoretical calculations. By tailoring the substrate composition-and therefore its lattice parameter-a compressive strain as high as 2.4 per cent is applied to the epitaxial α-FAPbI3 thin film. We demonstrate that this strain effectively changes the crystal structure, reduces the bandgap and increases the hole mobility of α-FAPbI3. Strained epitaxy is also shown to have a substantial stabilization effect on the α-FAPbI3 phase owing to the synergistic effects of epitaxial stabilization and strain neutralization. As an example, strain engineering is applied to enhance the performance of an α-FAPbI3-based photodetector.