identity theory
The identity theory (IT) of mind is standardly understood to be the claim that every mental property is identical with some physical property.

Identity theories of mind (IT) and multiple realization
The version of physicalism that gained ground in the middle of this century was in the first place committed to the identity thesis, namely, that every property (and thereby every mental property) is identical with some physical property.
IT: Every property is identical with some physical property.
Smart's 'Sensations and Brain Processes' (1959) provides the classic statement of this view. At that point, the primary concern of physicalists was to establish the point that two nonsynonymous terms could nonetheless pick out the same property. Even though 'pain' and, say, 'C-fibers firing,' are not synonymous, the property of being in pain could be identical with the property of having one's C-fibers firing. Very soon, however, the predominant concern shifted to what is now famous as the problem of multiple realizability and its implications for identity theories.
It's important to distinguish the problem of multiple realizability from the justification of multiple realizability. The justification stems from functionalism. According to functionalism, each mental property can be defined as a second-order property, the property of having some property or other that plays a certain functional role, defined in terms of other functional properties and physical causes and effects. As a result, it is likely that many different first-order physical properties can play the same functional role. There will (likely) be a very large set of different physical properties such that each can play the functional role, so that different instances of M may be correlated with various members of that set.
The way in which this constitutes a problem for identity theories is by implying a failure of coextension. Call this the Coextension Argument:
  1. There exists at least one mental property M such that there is a set of distinct physical properties {P1 v P2 v ... Pn}, no one of which realizes M on every occasion, while each realizes it on some occasion.
  2. If M is identical with any physical property, it is identical with one of {P1 v P2 v ... Pn}.
  3. M can't be identical with any of {P1 v P2 v ... Pn} because there is no member of it with which it is coextensive.
  4. There is no physical property with which M is coextensive, and hence none with which it is identical. 
Therefore, IT is false. A property M exists which is not identical to any physical property.
The identity theory IT can be saved in light of this argument by rejecting either premise 1 or premise 2. The disjunction option and elimination options discussed below correspond to those moves. If these prove unpalatable, the physicalist can still propose some theses that maintain the spirit, if not the letter, of IT; these are trope theories and second-order definition theories. I address each option in turn.
The disjunction option
One could deny 2 by finding some further physical property, unrelated to the realizers, which that is indeed in common to all the instances of M. But there is no reason to expect there to be any such property. The disjunction option instead locates the common property as the property defined by disjoining the members of {P1, P2, ... Pn}. The property M is identical with a physical property after all, namely, P1 v P2 v ... Pn.
Advocates of the disjunction option face two important questions. First, is P1 v P2 v ... Pn a property? Second, is it in fact identical with M? Many philosophers are suspicious of such disjunctive properties as that allegedly named by 'P1 v P2 v ... Pn' (Armstrong 1978; Lewis 1983). If one believes that every predicate corresponds to a property or a universal, then this problem cannot arise. But if you reject this presumption, then the question becomes pressing. (For debate, see Fodor 1974, Kim 1989, Kim 1992.)
Even if we decide that P1 v P2 v ... Pn is a genuine physical property, the question remains whether it can be identified with M. The original problem multiple realizability posed for IT was a failure of coextension, and P1 v P2 v ... Pn is supposed to be coextensive with M. But this is ambiguous: do we mean coextensive in the actual world, in all nomologically possible worlds, or all worlds whatsoever? If P1 v P2 v ... Pn is to be identical with M then, trivially, we must mean that they are coextensive in all worlds whatsoever. But, in fact, if we are tempted to the disjunctive option because of functionalism, we cannot appeal to a purely physical disjunction for an identity claim. It's a familiar point that functionalism not only allows variable physical realization; it also allows nonphysical realization. In some worlds, nonphysical properties will realize the mental. Hence, if P1 v P2 v ... Pn is to be coextensive with M in all possible worlds whatsoever, it must include some nonphysical properties as disjuncts. In that case, however, it is not a physical property with which we are identifying the physical property (Melnyk 1996).
The eliminative option
If one is an eliminativist about the mental in general, one is of course unfazed by the Coextension Argument. However, one could adopt a mild sort of eliminativism by denying, not that any mental properties exist at all, but by denying the more limited claim that is the first premise of the Coextension Argument.
The strategy is simple. For each of the members of {P1 v P2 v ... Pn} there is said to correspond a mental property, a species of M. If M is the property of pain, there will be many different kinds of pain, but no such thing as pain, period. Each of those kinds is then identified with its corresponding physical realizer, and IT is re-established.
Unlike a more thoroughgoing eliminativism, this view is not saddled with the implausible claim that people are not conscious, that they have no thoughts, etc. But it does face the charge that its invocation of these kinds of pain, and so on, is arbitrary and misleading. Why call it pain if doesn't have the features common to all pains? We cannot, on this view, say that it has those features, because there is no such thing as the property of pain, period, and hence no such thing as what is in common to each such instance. Perhaps, again invoking the distinction between predicates that correspond to properties and those that don't, the advocate of milder eliminativism could say that each such instance is picked out by the predicate 'pain,' and that is what they have in common; but this predicate fails to pick out any property (Kim 1992; Hooker 1981).
Identity theories and tropes
The remaining two responses to the Coextension Argument admit that IT is, in fact, false; they propose closely related theses instead, however, that maintain the basic idea of IT.
The first of these switches from an identity theory about properties to one about tropes. A trope is the particular instance of a property, an abstract particular. The Coextension Argument is powerless against this thesis:
IT1: Every actual trope is identical with a physical trope.
Even if there are properties -- universals -- that fail to be coextensive with physical properties, each instance of a nonphysical property could be identical with an instance of a physical property. If that were the case, physicalist intuitions would, it seems, be assuaged.
It's important to see that IT1 is more than a token identity thesis. It's not just that all particular events, or all particular objects, are physical; it's that all particular property instances are physical (Robb 1997).
The primary difficulty with this approach is simply that the nature of tropes is obscure and, as a result, it is nearly impossible to evaluate the claim that we have in certain cases not two tropes but one. Property individuation is puzzling enough; trope individuation is worse yet.
The second-order option
Finally, if functionalism is adopted, one has already provided an identity theory of a sort. That is, one has already subscribed to the following thesis:
IT2: Every property is identical with a physical property or a second-order property defined in purely physical and logical terms.
IT2 is not, of course, an identity theory in the standard sense. It does not imply that every property is a physical property, unless every physically definable second order property thereby counts as a physical property.
One problem with IT2 is that it requires that one buy into functionalism for every property for which multiple realizability is a problem. This may seem innocuous enough if one is convinced of multiple realizability because of a commitment to functionalism in the first place, but IT2 highlights how contentious a thesis functionalism is in the first place. It is fair to call it a sort of 'reductionism,' since it claims that every property can be defined using only logical apparatus and physical terms. If IT2 is one's method of saving the spirit of the identity theory, one certainly doesn't dilute its strength at all (Field 1992).
Increasingly, physicalists have appealed to a related thesis:
IT3: Every property is identical with a physical property or is realized on every occasion by a physical property.
(See Papineau 1993; Melnyk 1996.) Depending on exactly how the notion of realization is cashed out, this may turn out to be identical with IT2. The notion of realization has, however, been neglected in the philosophical literature, even as its appeal has grown.
Gene Witmer