In many ways it would be very desirable if the only concerns relevant to evaluating arguments were purely formal -- it would simplify things considerably. However, there are many problems that suggest that, even when we are considering rigorously constructed arguments, we cannot focus on the formal alone. One of these, and the best known, is equivocation, which is non-formal but has a massive influence on the validity of arguments -- an argument can only be valid on the presupposition that it does not equivocate. Another, and in some ways equally interesting, problem is begging the question, or circular argument.
It's very clear that we need some way of recognizing whether an argument begs the question, since not committing the fallacy is an important condition for application of an argument to reality. What isn't clear is how to pin the fallacy down. Aristotle, for instance, lists five ways in which an argument can beg the point at hand, and his treatment of the fallacy differs somewhat depending on whether he is discussing it in an epistemic context (e.g., when discussing demonstration) or in a dialectical context (e.g., in the Topics). However, I think Aristotle's analysis of the fallacy is extremely interesting, and worth keeping in mind.
According to Aristotle, begging the question is a failure of demonstration. On his view, demonstration has an epistemic requirement: the premises have to be at least as well known as the conclusion. It is this that gives a demonstration its superiority over other deductive arguments, and it is this that makes demonstration so useful: when we demonstrate we advance in knowledge by strengthening the certainty of the conclusion. In a circular argument, we fail to do this (not every violation of the epistemic requirement is a case of circularity, of course).
However, while all arguments that beg the question are circular arguments, not all circular arguments beg the question. There are perfectly legitimate reasons, of course, why one might construct a circular argument, as an exercise, for instance. But Aristotle has something different in mind. When we argue A -> A or A -> B -> A, there is at least one type of case where this is clearly not a problem: the case in which A is self-evident. It is, Aristotle thinks, impossible to beg the question with self-evident conclusions. This leads Aristotle to the interesting, and I think very insightful, point that if we allowed all cases of begging the question, we would effectively be treating all conclusions as self-evident. In such a situation, of course, logical argument becomes otiose.
Things become more complicated in a dialectical context, because we are now involved in a dialogue with more than one person, and the questions of whether a premise is recognizably as well known as its conclusion, and whether an argument recognizably begs the question become more important.
So reflecting on the fallacy of begging the question gets us into interesting epistemic and dialectical issues. Of course, there still is some interest in seeing whether a purely formal account of begging the question is possible. The best known, and most interesting, attempt to sketch such an account is that of De Morgan. J. S. Mill had argued that all deductive arguments are circular in virtue of their form. Of course, under Aristotle's analysis, this objection fails, since it's beaten by the epistemic requirement. De Morgan further pointed out that Mill misunderstood the nature of the minor premise. But he also went and suggested a more formal test for determining whether an argument does not commit the fallacy: No deductive argument begs the question if it has more than one premise and no superfluous premises. It isn't quite clear whether this test is entirely effective; Woods and Walton*, for instance, note that Aristotle still seems to be right that epistemic and dialectical considerations are necessary to identify the fallacy correctly in all cases.
* Douglas Walton and James Woods, "The Petitio: Aristotle's Five Ways," Canadian Journal of Philosophy 12 (1982) 77-100.
