In this paper I argue that it is possible for someone to intend and to do something that he believes to be impossible. The argument proceeds by appeal to thought experiments in which intuitively someone does an action of a certain type intentionally, for example, move his arm, though he believes it to be impossible. I then invoke the so-called Simple View of the connection between intending and doing intentionally according to which if one does A intentionally then one intended to A to arrive at the conclusion that it is possible to intend to do what one believes to be impossible. I defend the Simple View against some recent attacks, notably by Michael Bratman, and explain how it is possible for one to intend to do something which one nevertheless believes cannot be done. The possibility turns on our being able to find a reason to do something that aims for an end that is not provided by the value that the probability of success in achieving the end has for us. The simplest sort of example involves someone’s trying to do something in order to show that it is, as he thinks, impossible, though his belief is mistaken: when he succeeds, he has certainly done intentionally the thing that he thought impossible, and likewise, if the Simple View is correct, intended to do it. An immediate consequence is that a standard view of the nature of action explanation, according to which nothing is an action unless it is “rationalized” by a belief/desire pair in the light of whose contents it is seen to have something to be said in its favor, misrepresents the complexity of agency and must be revised.