The study of Turing machines as dynamical systems carried several questions that were not considered before, as the question of the existence of an << aperiodicity >>, finally solved by the very particular machine called SMART.

Combined with other techniques, this machine opened the door for a series of new findings.

In this talk we will recall some of the highlights of this story, but also show some of the remaining open problems, together with some of the attempts to solve them.