Summary: We prove that the Benjamin-Ono equation \[ \partial_t u+ H\partial^2_x u+ u\partial_x u= 0 \] is locally well-posed in \(H^s(\mathbb{R})\) for \(s> 9/8\) and that for arbitrary initial data, the modified (cubic nonlinearity) Benjamin-Ono equation \[ \partial_t u+ H\partial^2_x u+ u^2\partial_x u= 0 \] is locally well-posed in \(H^s(\mathbb{R})\) for \(s\geq 1\).