The author considers the initial value problem for the forced KdV equation \[ \begin{cases} \partial _t u + \partial _x^3 u + u\partial _x u = f, \quad (x,t) \in \mathbb R\times[0,\infty ), \\ u(x,0) = u_0 (x) \in H^s(\mathbb R), \end{cases} \] where \(u(x,t)\) and \(f(x)\) are real valued functions. The time local well-posedness with \(f(x) \in H^\sigma\), \(\sigma \geq - 3\) and the time global well-posedness with \(f = p\delta^{'}(x)\) or\(f(x) \in H^\sigma\), \(\sigma \geq - 3/2\) are studied. The main tools are the Fourier restriction norm method and \(I\)-method.