The title of this talk is “Your Turing Machine or Mine?”. What I am alluding to with this title is the universality of a universal Turing machine, and at the same time, to the fact that there are many different universal Turing machines with somewhat different properties. Universal Turing machines are both universal and individual, in different senses. A universal Turing machine is one that can emulate (or imitate) any other Turing machine, and thus in a sense can undertake to compute any of a very large class of computable functions. But there are an indefinitely large number of universal Turing machines that can be defined.