The ASCA satellite made 107 pointing observations on a 5* 5 deg2 region around the center of our Milky Way Galaxy (the Galactic Center) from 1993 to 1999. In the X-ray images of the 0.7-3 keV or 3-10 keV bands, we found 52 point sources and a dozen diffuse sources. All the point sources are uniformly fitted with an absorbed power-law model. For selected bright sources, Sgr A^*, AX J1745.6-2901, A 1742-294, SLX 1744-300, GRO J1744-28, SLX 1737-282, GRS 1734-292, AX J1749.2-2725, KS 1741-293, GRS 1741.9-2853, and an unusual flare source XTE J1739-302, we present further detailed spectral and timing analyses, and discuss on their nature. A dozen extended X-ray sources comprise radio supernova remnants, giant molecular clouds, and those new discoveries. Most show emission lines from either highly ionized atoms or low-ionized irons. The X-ray spectra were fitted with either a thin thermal or power-law model. This paper summarizes the results and provides the ASCA X-ray source catalogue in the Galactic Center region.