Superconducting ${\mathrm{MgB}}_{\mathrm{2}}$ thin films were prepared on ${\mathrm{Al}}_{\mathrm{2}}{\mathrm{O}}_{\mathrm{3}}\mathrm{(0001)}$ and MgO(100) substrates. Boron thin films were deposited by the electron-beam evaporation followed by postannealing process with magnesium. Proper postannealing conditions were investigated to grow good superconducting ${\mathrm{MgB}}_{\mathrm{2}}$ thin films. The x-ray diffraction patterns showed randomly oriented growth of ${\mathrm{MgB}}_{\mathrm{2}}$ phase in our thin films. The surface morphology was examined by scanning electron microscope and atomic force microscope. Critical current density ${\mathrm{(J}}_C)$ measured by transport method was about ${10}^7 {\mathrm{A/cm}}^{\mathrm{2}}$ at 15 K, and superconducting transition temperature ${\mathrm{(T}}_C)$ was ∼39 K in the ${\mathrm{MgB}}_{\mathrm{2}}$ thin films on ${\mathrm{Al}}_{\mathrm{2}}{\mathrm{O}}_{\mathrm{3}}.$