  Simultaneous diagonal flips in plane triangulations are investigated. It is
proved that every $n$-vertex triangulation with at least six vertices has a
simultaneous flip into a 4-connected triangulation, and that it can be computed
in O(n) time. It follows that every triangulation has a simultaneous flip into
a Hamiltonian triangulation. This result is used to prove that for any two
$n$-vertex triangulations, there exists a sequence of $O(\log n)$ simultaneous
flips to transform one into the other. The total number of edges flipped in
this sequence is O(n). The maximum size of a simultaneous flip is then studied.
It is proved that every triangulation has a simultaneous flip of at least
${1/3}(n-2)$ edges. On the other hand, every simultaneous flip has at most
$n-2$ edges, and there exist triangulations with a maximum simultaneous flip of
${6/7}(n-2)$ edges.
