The epitaxial growth of $CeO_2$ films has been investigated on three different substrates-Si(111), $SrTiO_3$(001), and MgO(001)-over wide range of growth parameters using oxygen-plasma-assisted molecular beam epitaxy. Pure-phase, single-crystalline epitaxial films of $CeO_2$ (001) have been grown only on $SrTiO_3$(001). We discuss the growth conditions in conjunction with the choice of substrates required to synthe-size this oxide, as well as the associated characterization by menas of x-ray diffraction, reflection high-energy electron diffraction, low-energy electron diffraction, and x-ray photoelectron spectroscopy and diffraction. Successful growth of single crystalline $CeO_2$ depends critically on the choice of substrate and is rather insensitive to the growth conditions studied in this investigation. $CeO_2$(001) films on $SrTiO_3$exhibit the sturcture of bulk $CeO_2$ without surface reconstructions. Ti outdiffusion is observed on the films grown temperatures above $650^{circ}C$.