We have grown AlN nanorods and AlN films using plasma-assisted molecular beam epitaxy by changing the Al source flux. Plasma-assisted molecular beam epitaxy of AlN was performed on c-plane $Al_2O_3$ substrates with different levels of aluminum (Al) flux but with the same nitrogen flux. Growth behavior of AlN was strongly affected by Al flux, as determined by in-situ reflection high energy electron diffraction. Prior to the growth, nitridation of the $Al_2O_3$ substrate was performed and a two-dimensionally grown AlN layer was formed by the nitridation process, in which the epitaxial relationship was determined to be [11-20]AlN//[10-10]$Al_2O_3$, and [10-10]AlN//[11-20]$Al_2O_3$. In the growth of AlN films after nitridation, vertically aligned nanorod-structured AlN was grown with a growth rate of $1.6{mu}m/h$, in which the growth direction was <0001>, for low Al flux. However, with high Al flux, Al droplets with diameters of about $8{mu}m$ were found, which implies an Al-rich growth environment. With moderate Al flux conditions, epitaxial AlN films were grown. Growth was maintained in two-dimensional or three-dimensional growth mode depending on the Al flux during the growth; however, final growth occurred in three-dimensional growth mode. A lowest root mean square roughness of 0.6 nm (for $2{mu}m{ imes}2{mu}m$ area) was obtained, which indicates a very flat surface.