Applicability of Church-Turing thesis to interactive models of computation Paul Wegner and Dina Goldin have for over a decade been publishing papers and books arguing primarily that the Church-Turing thesis is often misrepresented in the CS Theory community and elsewhere.  That is, it is presented as encompassing all computation when in fact it applies only to computation of functions, which is a very small subset of all computation.  Instead they suggest we should seek to model interactive computation, where communication with the outside world happens during the computation.

The only critique I have seen of this work is in the Lambda the Ultimate forum, where somebody lamented these authors for continually publishing what is obviously known.  My question then is, is there any more critique into this line of thinking, and in particular their Persistent Turing Machines.  And if not, why then is it seemingly studied very little (I could be mistaken).  Lastly, how does the notion of universality translate to an interactive domain.
Here's my favorite analogy.  Suppose I spent a decade publishing books and papers arguing that, contrary to theoretical computer science's dogma, the Church-Turing Thesis fails to capture all of computation, because Turing machines can't toast bread.  Therefore, you need my revolutionary new model, the Toaster-Enhanced Turing Machine (TETM), which allows bread as a possible input and includes toasting it as a primitive operation.

You might say: sure, I have a "point", but it's a totally uninteresting one.  No one ever claimed that a Turing machine could handle every possible interaction with the external world, without first hooking it up to suitable peripherals.  If you want a TM to toast bread, you need to connect it to a toaster; then the TM can easily handle the toaster's internal logic (unless this particular toaster requires solving the halting problem or something like that to determine how brown the bread should be!).  In exactly the same way, if you want a TM to handle interactive communication, then you need to hook it up to suitable communication devices, as Neel discussed in his answer.  In neither case are we saying anything that wouldn't have been obvious to Turing himself.

So, I'd say the reason why there's been no "followup" to Wegner and Goldin's diatribes is that TCS has known how to model interactivity whenever needed, and has happily done so, since the very beginning of the field.

Update (8/30): A related point is as follows.  Does it ever give the critics pause that, here inside the Elite Church-Turing Ivory Tower (the ECTIT), the major research themes for the past two decades have included interactive proofs, multiparty cryptographic protocols, codes for interactive communication, asynchronous protocols for routing, consensus, rumor-spreading, leader-election, etc., and the price of anarchy in economic networks? If putting Turing's notion of computation at the center of the field makes it so hard to discuss interaction, how is it that so few of us have noticed?

Another Update: To the people who keep banging the drum about higher-level formalisms being vastly more intuitive than TMs, and no one thinking in terms of TMs as a practical matter, let me ask an extremely simple question.  What is it that lets all those high-level languages exist in the first place, that ensures they can always be compiled down to machine code?  Could it be ... err ... THE CHURCH-TURING THESIS, the very same one you've been ragging on?  To clarify, the Church-Turing Thesis is not the claim that "TURING MACHINEZ RULE!!"  Rather, it's the claim that any reasonable programming language will be equivalent in expressive power to Turing machines -- and as a consequence, that you might as well think in terms of the higher-level languages if it's more convenient to do so.  This, of course, was a radical new insight 60-75 years ago.

Final Update: I've created a blog post for further discussion of this answer.