In most Germanic and Romance languages the present perfect has developed from a resultative meaning via an anterior into absolute past. In Functional Discourse Grammar terms this corresponds to the grammaticalization of a phasal aspectual operator at the layer of the Configurational Property, via a relative tense operator at the layer of the State-of-Affairs, into an absolute tense operator at the layer of the Episode. This is what happened in Romance languages, such as French and Italian, while Peninsular Spanish is developing in the same direction, without as yet having fully reached the absolute past stage. The Portuguese present perfect, however, is different as it does not express resultative aspect, relative past or absolute past meaning but rather the iteration or continuity of an event from some past moment onward until after the moment of speaking. A further idiosyncrasy of the perfect in Portuguese is that the auxiliary is based on Latin tenere rather than habere, as is the case in the other Romance languages. This paper describes the semantic and the morphosyntactic aspects of the grammaticalization of the (Brazilian) Portuguese perfect in diachrony and synchrony. It turns out that (i) the medieval habere-based Portuguese present perfect becomes obsolete and the past perfect develops into a relative past, (ii) the post-medieval tenere-based past perfect turns into a relative past as well, whereas (iii) the tenere-based present perfect undergoes semantic specialization in the course of the 20th century. This paper shows how these facts can be accounted for within the Functional Discourse Grammar approach to the grammaticalization of aspect and tense.