What goes on when one person forgives another? In this paper I argue for The Alteration Thesis: that forgiveness alters the normative situation created by wrongdoing. Furthermore, I argue that it does so by means of the exercise of a normative power. I also argue that there are two main forms of such forgiveness: rights-waiving and redemptive. While forgiveness may – but does not always – alter obligations by waiving them, I claim that it also alters the normative situation by creating a new obligation to the wrongdoer. Thinking of forgiveness along the lines suggested by the Alteration Thesis means going against the tide of much recent writing on forgiveness, which has seen forgiveness as consisting essentially in a change of heart towards the wrongdoer. But I argue that the Alteration Thesis has a number of explanatory advantages over the Change of Heart approach.