Here we report the growth of sub-millimeter MgB2 single crystals of various shapes under high pressure in Mg-B-N system. Structure refinement using a single-crystal X-ray diffraction analysis gives lattice parameters a=3.0851(5) A and c=3.5201(5) A with small reliability factors (Rw =0.025, R=0.018), which enables us to analyze the Fourier and Fourier difference maps. The maps clearly show the B sp2 orbitals and covalency of the B-B bonds. The sharp superconducting transitions at Tc =38.1-38.3K were obtained in both magnetization (DTc =0.6K) and resistivity (DTc <0.3K) measurements. Resistivity measurements with magnetic fields applied parallel and perpendicular to the Mg and B sheets reveal the anisotropic nature of this compound, with upper critical field anisotropy ratio of about 2.7.