Why one-dimensional strings, but not higher-dimensional shells/membranes? One way that I've seen to sort-of motivate string theory is to 'generalize' the relativistic point particle action, resulting in the Nambu-Goto action.  However, once you see how to make this 'generalization', it becomes obvious how to write down the action, not just for a string, but for manifolds of higher dimension as well.  In fact, Becker-Becker-Schwarz (the main source I happen to be learning from) actually do this.  But (as far as I have read), they merely write down the action and do nothing further with it.
My question is:  what happens when we proceed along the same lines as string theory, but when replacing a string with a 2-manifold, the simplest example of which would be the 2-sphere, a "shell/membrane"?  Assuming 3 spatial dimensions, this is the highest dimensional manifold we can consider (because we don't allow for non-compact manifolds).  Furthermore, there is only one compact manifold of dimension 1; however, there are infinitely many compact manifolds of dimension 2, which could potentially make the theory much richer (and probably much more difficult).  For example, in string theory, we are stuck with $S^1$ (if you insist upon having no boundaries), but if you allow for 2 dimensions, we could consider the sphere, the torus, etc.
Because it seems that this path is not ever presented (in fact, I've never even heard of it), I would presume that something goes wrong.  So then, what exactly does go wrong?
 A: Maybe it should be remembered that the late 90s not only saw M(atrix) speculation, but also the observation that most, if not all, higher branes in string/M-theory, at least as far as their "worldvolume theory" is concerned, have a quantum description in terms of AdS-CFT duality.
Notably the quantum M2-brane (which is the super-membrane in 11-dimensional spacetime that is mentioned elsewhere in this thread) is supposed to be described by AdS4-CFT3 duality. Similarly the quantum M5-brane is supposed to be given by AdS7-CFT6, and so forth.
In this context, here is something maybe to keep in mind: according to (Witten 98) as far as the conformal blocks on the CFT-side are concerned, AdS/CFT is all controled (just) by the higher dimensional Chern-Simons sector inside the higher dimensional supergravity theory. For instance by the 7-dimensional Chern-Simons term that remains from the 11-dimensional supergravity CS term after compactification on the 4-sphere. Moreover, under this identification the CFT on the worldvolume of the brane is supposed to be related to the higher CS-theory in direct analogy to how the ordinary WZW model is related to ordinary 3d Chern-Simons theory under the seminal CS-WZW correspondence.
So by this Chern-Simons/WZW-perspective on AdS/CFT of (Witten 98) one may expect that all branes have holographic quantum descriptions as "higher dimensional WZW models". But ever since (Henneaux-Mezincescu 85) is it "sort of clear" that the Green-Schwarz sigma-model description of all or most of the super $p$-branes of string/M-theory are of "higher WZW type" in somse sense. 
With coauthors we have recently expanded a bit on this perspective in arXiv:1308.5264, showing that indeed all the super $p$-branes in particular also those containing tensor multiplet fields (such as the M5 and the D$p$s) are in a systematic and precise sense higher WZW theories which are boundary theories of higher Chern-Simons type theories. 
Daniel Freed has recently been re--amplifying this kind of perspective in view of the cobordism hypothesis theorem, see 


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*Daniel Freed, 4-3-2 8-7-6, talk at ASPECTS of Topology, Oxford Dec 2012 (pdf)


I guess in conclusion I am just trying to say: ever since AdS/CFT we do know a good bit more about the quantization of all or at least many of the higher dimensional branes in string/M-theory. At least in principle. Stuff remains to be worked out.
A: A classical1  theory of (relativistic) $p$-dimensional membranes exists for any non-negative integer $p$. Such classical membrane-like objects appears in the full non-perturbative formulation of string theory. 
The problem arises if one tries to quantize a membrane (in a first quantized sense) using standard perturbative quantization methods, which have been successfully applied to $0$-dimensional point-particles and $1$-dimensional strings. This program has so far failed for membranes with internal dimension $p\geq 2$. See e.g. Ref. 1 and Ref. 2. Intuitively, the problem is that membranes can grow spikes/tubes that don't cost energy. 
Nicolai et. al. have later given a second-quantized interpretation of membranes inside M(atrix) theory. See e.g. Ref. 3. 
References:


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*B. de Wit, J. Hoppe, and H. Nicolai. On the Quantum Mechanics of
Supermembranes, Nucl. Phys. B305 (1988) 545.

*B. de Wit, W. Lüscher, and H. Nicolai, The supermembrane is unstable, Nucl. Phys. B320 (1989) 135. 

*H. Nicolai and R. Helling, Supermembranes and M(atrix) Theory,  Lectures at Trieste Spring School (1998). (Hat tip: alexarvanitakis.)

1      Here the word classical means $\hbar=0$.
