Triquotient maps of topological spaces were introduced by E. Michael as a natural generalization of both open and proper surjections. We introduce the notion of localic triquotient map. Our main result is that localic triquotient maps are effective descent maps. This generalizes the corresponding results for proper surjections (J. Vermeulen) and open surjections (A. Joyal and M. Tierney). Further results concern stability of triquotiency under various operations, for instance, arbitrary products and filtered (inverse) limits. Among the applications are a new constructive proof of Tychonoff's theorem, and a new result on stability of open surjections under filtered limits.