  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.
