  Superconducting thin films of magnesium diboride (MgB$_2$) were prepared on
MgO (001) substrate by a molecular beam epitaxy (MBE) method with the
co-evaporation conditions of low deposition rate in ultra-high vacuum. The
structural and physical properties of the films were studied by RHEED, XRD,
XPS, resistivity and magnetization measurements.All films demonstrated
superconductivity without use of any post-annealing process.The highest {\it
T}$_{c,onset}$ determined by resistivity measurement was about 33K in the
present samples.Anisotropic superconducting properties were evaluated by the
resistivity and magnetic measurements.We will discuss the anisotropy of
superconductivity for as-grown MgB$_2$ thin films.
