Discrete Transfinite Computation

We describe various computational models based initially, but not exclusively, on that of the Turing machine, that are generalized to allow for transfinitely many computational steps. Variants of such machines are considered that have longer tapes than the standard model, or that work on ordinals rather than numbers. We outline the connections between such models and the older theories of recursion in higher types, generalized recursion theory, and recursion on ordinals such as $\alpha$-recursion. We conclude that, in particular, polynomial time computation on $\omega$-strings is well modelled by several convergent conceptions.