This article is concerned with the study of approximate controllability for the semilinear heat equation in a bounded domain Ω when the control acts on any open and nonempty subset of Ω or on a part of the boundary. In the case of both an internal and a boundary control, the approximate controllability in L P (Ω) for 1 ≦ p < + ∞ is proved when the nonlinearity is globally Lipschitz with a control in L ∞ . In the case of the interior control, we also prove approximate controllability in C 0 (Ω). The proof combines a variational approach to the controllability problem for linear equations and a fixed point method. We also prove that the control can be taken to be of “quasi bang-bang” form.