A classical result of Wagner states that any two (unlabelled) planar triangulations with the same number of vertices can be transformed into each other by a finite sequence of diagonal flips. Recently Komuro gives a linear bound on the maximum number of diagonal flips needed for such a transformation. In this paper we show that any two labelled triangulations can be transformed into each other using at most O(n log n) diagonal flips. We also show that for planar triangulations a linear number

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... flips suffice. We will also show that any planar triangulation with n > 4 vertices has at least n − 2 flippable edges. Finally, we prove that if the minimum degree of a triangulation is at least 4 then it contains at least 2n + 3 flippable edges. These bounds are tight.