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The epitaxial growth of technically-important $\beta$-Ga$_2$O$_3$ semiconductor thin films have not been realized on flexible substrates due to limitations by the high-temperature crystallization conditions and the lattice-matching requirements. In this report, for the first time single crystal $\beta$-Ga$_2$O$_3$(-201) thin films is epitaxially grown on the flexible CeO2 (001)-buffered hastelloy tape. The results indicate that CeO$_2$ (001) has a small bi-axial lattice mismatch with $\beta$-Ga$_2$O$_3$ (-201), thus inducing a simultaneous double-domain epitaxial growth. Flexible photodetectors are fabricated based on the epitaxial $\beta$-Ga$_2$O$_3$ coated tapes. Measurements show that the obtained photodetectors have a responsivity of 40 mA/W, with an on/off ratio reaching 1000 under 250 nm incident light and 5 V bias voltage. Such photoelectrical performance is already within the mainstream level of the $\beta$-Ga$_2$O$_3$ based photodetectors by using the conventional rigid single crystal substrates; and more importantly remained robust against more than 1000 cycles of bending tests. In addition, the epitaxy technique described in the report also paves the way for the fabrication of a wide range of flexible epitaxial film devices that utilize the materials with lattice parameters similar to $\beta$-Ga$_2$O$_3$, including GaN, AlN and SiC., Comment: 25 pages, 10 figures