  We establish the local well-posedness of the generalized Benjamin-Ono
equation $\partial_tu+\mathcal{H}\partial_x^2u\pm u^k\partial_xu=0$ in
$H^s(\R)$, $s>1/2-1/k$ for $k\geq 12$ and without smallness assumption on the
initial data. The condition $s>1/2-1/k$ is known to be sharp since the solution
map $u_0\mapsto u$ is not of class $\mathcal{C}^{k+1}$ on $H^s(\R)$ for
$s<1/2-1/k$. On the other hand, in the particular case of the cubic
Benjamin-Ono equation, we prove the ill-posedness in $H^s(\R)$, $s<1/3$.
