Strain engineering is a powerful tool with which to enhance semiconductor device performance(1,2). 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 pressurization(6-8), electrostriction(9), annealing(10-12), van der Waals force(13), thermal expansion mismatch(14), and heat-induced substrate phase transition(15), 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 a-formamidinium lead iodide (alpha-FAPbI(3)) 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 alpha-FAPbI(3) thin film. We demonstrate that this strain effectively changes the crystal structure, reduces the bandgap and increases the hole mobility of alpha-FAPbI(3). Strained epitaxy is also shown to have a substantial stabilization effect on the alpha-FAPbI(3) 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 alpha-FAPbI(3)-based photodetector.