Taking Per Martin-Löf’s constructive type theory as a starting-point a theory of assertion is developed, which is able to account for the epistemic aspects of the speech act of assertion, and in which it is shown that assertion is not a wide genus. From a constructivist point of view, one is entitled to assert, for example, that a proposition A is true, only if one has constructed a proof object a for A in an act of demonstration. One thereby has grounded the assertion by an act of demonstration, and a grounding account of assertion therefore suits constructive type theory. Because the act of demonstration in which such a proof object is constructed results in knowledge that A is true, the constructivist account of assertion has to ward off some of the criticism directed against knowledge accounts of assertion. It is especially the internal relation between a judgement being grounded and its being known that makes it possible to do so. The grounding account of assertion can be considered as a justification account of assertion, but it also differs from justification accounts recently proposed, namely in the treatment of selfless assertions, that is, assertions which are grounded, but are not accompanied by belief.