We consider bilinear optimal control problems whose objective functionals do not depend on the controls. Hence, bang-bang solutions will appear. We investigate sufficient second-order conditions for bang-bang controls, which guarantee local quadratic growth of the objective functional in L-1. In addition, we prove that for controls that are not bang-bang, no such growth can be expected. Finally, we study the finite-element discretization and prove error estimates of bang-bang controls in L-1-norms.