We report plasma-assisted molecular beam epitaxy of $In_XGa_{1-X}N$ films on c-plane sapphire substrates. Prior to the growth of $In_XGa_{1-X}N$ films, GaN film was grown on the nitride c-plane sapphire substrate by two-dimensional (2D) growth mode. For the growth of GaN, Ga flux of $3.7{ imes}10^{-8}$ torr as a beam equivalent pressure (BEP) and a plasma power of 150 W with a nitrogen flow rate of 0.76 sccm were fixed. The growth of 2D GaN growth was confirmed by $in-situ$ reflection high-energy electron diffraction (RHEED) by observing a streaky RHEED pattern with a strong specular spot. InN films showed lower growth rates even with the same growth conditions (same growth temperature, same plasma condition, and same BEP value of III element) than those of GaN films. It was observed that the growth rate of GaN is 1.7 times higher than that of InN, which is probably caused by the higher vapor pressure of In. For the growth of $In_xGa_{1-x}N$ films with different In compositions, total III-element flux (Ga plus In BEPs) was set to $3.7{ imes}10^{-8}$ torr, which was the BEP value for the 2D growth of GaN. The In compositions of the $In_xGa_{1-x}N$ films were determined to be 28, 41, 45, and 53% based on the peak position of (0002) reflection in x-ray ${ heta}-2{ heta}$ measurements. The growth of $In_xGa_{1-x}N$ films did not show a streaky RHEED pattern but showed spotty patterns with weak streaky lines. This means that the net sticking coefficients of In and Ga, considered based on the growth rates of GaN and InN, are not the only factor governing the growth mode; another factor such as migration velocity should be considered. The sample with an In composition of 41% showed the lowest full width at half maximum value of 0.20 degree from the x-ray (0002) omega rocking curve measurements and the lowest root mean square roughness value of 0.71 nm.