Effects of substrates and buffer layer upon the formation of crystalline phases and ferroelectricity of $YMnO_3$ thin films were investigated. The hexagonal $YMnO_3$ was easily formed on Si(100) while the mixed phases, hexagonal and orthorhombic $YMnO_3$, on $Pt(111)/TiO_2/SiO_2/Si$ substrate. When the $Y_2O_3$ buffer layer of 70 nm thick was inserted between the substrates and the $YMnO_3,$ the c-axis oriented hexagonal single phase formed on both substrates, Si(100) and $Pt(111)/TiO_2/SiO_2/Si$. The leakage current density of the hexagonal $YMnO_3$ thin films was lower than that consisting of mixed phases, hexagonal and orthorhombic. Furthermore the hexagonal $YMnO_3$ with c-axis preferred orientation showed the lowest leakage current density. The remnant polarization from a P-E hysteresis curve for the $YMnO_3$ formed on Si(100) was 0.14 without buffer layer and $0.24_{mu}C/cm^2$ for that with buffer layer. For the $Pt(111)/TiO_3/SiO_3/Si$ substrates, the specimen without $Y_2O_3$buffer layer did not show the hysteresis curve, while the buffer-layered has the remnant polarization of $1.14_{mu}C/cm^2$. It was concluded that the leakage current density and the ferroelectricity for the $YMnO_3$ thin films could be controlled by varying crystalline phases and their preferred orientation which depend on the kind of substrates and whether the $Y_2O_3$buffer layer exist or not.