To a significant degree the mind/body problem is a technical issue internal to
academic philosophy, though it has complex relations to wider
scientific and
philosophical issues. It has become fashionable to say that
physicalism solved the problem by showing how the mind is connected to -- or
is -- the
brain, and nervous system in general. This has been increasingly common among professional
philosophers as
cognitive science has taken off, and has always been common among
neuroscientists,
psychologists and other's with scientific leanings (though perhaps less so than one would suspect). In some respect this is true: science has provided overwhelming evidence that the mind and brain are one and the same, and found little evidence that any of our psychology inhabits a spiritual plane of existence. It seems natural to say the problem has been resolved and that we should move on to more important issues if only those pesky philosophers would acknowledge that the problem is an illusion.
The reason why solution hasn't been accepted by everyone in the philosophy community is simple: it doesn't address the mind that the critics believe needs explaining. There are very few philosophers who doubt that psychology is firmly grounded in the brain; almost no one seriously proposes that reason is the exclusive property of an immaterial soul anymore. They do claim, however, the physicalism, for all it's success, it silent on the issue of consciousness of the what-it-is-like sort (as opposed to wakefulness or attention, where science is doing quite well). In many respects this is an epistemological problem (the 'explanatory gap'): consciousness doesn't fit in physics because physics is designed to work with the objective world and can't provide us with information about the subjective elements that may exist. Perhaps there's no particular reason why consciousness couldn't be physical, but at this point we have no data on where in physics it could be. (Quantum mechanics -- vanilla QM; none of this revisionist stuff -- includes quite a bit of epistemology and other interesting philosophical issues, but it tends to beg questions, not answer them.) Solving this problem would take care of the 'where' and probably the 'when' of the mind/body problem once and for all, but leave the troubling 'how' and 'what'. ('Why' can go to your choice of theologians, physicists, or biologists.) Nevertheless, it would go a long way toward an eventual solution (or at any rate, cease-fire).
At some point epistemology gives way to metaphysics, which greets the problem with a quizzical look and vanishes into the study for several centuries to ponder just what 'experience' is. Among those metaphysicians who continue to believe that the M/BP is a problem -- which is to say, all who take their job seriously -- it seems to be largely agreed upon that materialism is wrong in some important way, even if no one can say just how. Dualism has never been a satisfactory answer simply because it isn't an answer, just a restatement of the problem. Idealism is better, but fares poorly in most other respects and once again doesn't offer much to explain consciousness. Recognition of the need for some sort of objective (insofar as the entire world wouldn't disappear in a puff of qualia if there were no observers) monism is nearly universal: there's no evidence the world isn't monistic and somewhat objective, and it makes things a lot easier if you don't have to work an interaction between worlds into the equation. Unfortunately, you are left a problem almost as large: how can something be both objective and subjective at the same time, and what do those two properties have in common that allows us to say that they are a unity? (Notice that we're on our way back to epistemology.) It has been suggested that this problem may be insoluble: not only may we not have the tools (intellectual and physical) needed to unravel the knot, but we may not be capable of understanding the answer if we found it.
Unfortunately, Tthe most promising result of all of this speculation is only that people have started paying attention to the problem again. The last few years have a torrent of activity in journals and conferences on consciousness and the M/BP (the most (in)famous being Tucson I-IV), and considerable interest from beyond philosophy (oddly, primarily from physicists; psych and neurosci still haven't come around). While things aren't looking up at all, it's certainly a much more exciting time than a few decades ago.
(Postscript: All of this is ignoring the looming problem of intentionality, which is superficially easier to explain but surprisingly difficult once you begin. Consciousness and intentionality have long been linked, and it's entirely possible that consciousness will never be satisfactorily described without first tackling intentionality. Thus far few people have even tried to bind the two together, and no proposal has escaped harsh criticism (that is, harsh criticism of actual problems, not the criticism that accompanies philosophy jounals like noise on my phone line [see US West]). Aside from the never ending debate over where meaning lives -- in the mind or in the world -- little progress has been made since the term reentered the English language. On the other hand, philosophy seems to have this one all to it's own; at least we don't have worry about those pesky scientists coming in and trying to explain away all our subject matter.)
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