Higher-dimensional Gromov-Witten theories A basic set-up in modern enumerative geometry is that you have some object $X$ (say, a "nice" stack, for whatever definition of "nice" you need) and then you want to "count" the curve of genus $g$ intersecting a bunch of cohomology classes and of degree $\beta$, so you look at $\bar{\mathcal{M}}_{g,n}(X,\beta)$, then pull back the classes, intersect, and get a Gromov-Witten Invariant.  Famously, this gives the Kontsevich formula counting rational curves in the projective plane passing through $3d-1$ points.  And though GW-invariants can be negative and rational, there are nice cases where they do count something legitimate, such as the genus $0$, $n\geq 3$ case into a homogeneous space.
So, enough background, here's my question (and this is largely idle curiosity, so no specific motivation): can we do this in higher dimensions? For instance, given a (smooth?) variety $V$ and marking a bunch of subvarieties $W_1,\ldots,W_n$ (maybe restricting them to points?) can we form $\bar{\mathcal{M}}_{V,(W_1,\ldots,W_n)}(X,\beta)$ a moduli space of stable mappings of varieties deformation equivalent to the one we started with into our space, represented by a given cohomology class $\beta$ and with $W_1,\ldots,W_n$ intersecting some cohomology classes, so that we can get something that can be called higher dimensional Gromov-Witten invariants? If this has been studied, under what conditions does it actually count subvarieties? For instance, if $X$ is $\mathbb{P}^N$ and $V=\mathbb{P}^2$, and maybe if we loosen things to just needing rational maps, could we use something like this to count rational surfaces, satisfying some incidence conditions?
 A: It seems pretty much impossible that a virtual fundamental class for a moduli space of maps from surfaces to a variety can be constructed with current techniques.
In the case of curves, deformations of a map are controlled by $H^0(C, f^* T_X)$, and obstructions by $H^1(C, f^* T_X)$. These two spaces vary nicely over the moduli space, in the sense that there is a two-term complex on the moduli space whose fiber at the point [C, f] has these cohomologies. (It is obtained by pulling back the tangent bundle to the universal curve, then taking its derived push-forward along the projection to the moduli space.)
If you do the same with surfaces, you end up with a 3-term complex. In that case, Behrend-Fantechi no longer help you to construct a virtual fundamental class. 
A: Alexeev's and Knutson's moduli space of branch varieties in

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*Complete moduli spaces of branchvarieties, Journal für die reine und angewandte Mathematik (Crelles Journal). Volume 2010, Issue 639, Pages 39-71, doi:10.1515/crelle.2010.010, arXiv:math/0602626
is kind of like a higher dimensional version of the moduli space of stable maps. (Although you should be warned that the moduli space of stable maps is not a special case of the moduli space of branch varieties. In the moduli space of branch varieties, there are never any maps with positive dimensional fibers.)
I don't know of anyone whose done serious enumerative work along these lines. I'm looking forward to reading the other answers.
A: The moduli space of stable maps to a point is the Deligne-Mumford moduli space of curves.  The moduli theory of higher-dimensional varieties is extremely complicated: though much has been done, even for surfaces there are many possible stability conditions.  So even the "higher Gromov-Witten theory of a point" would be very challenging to define.
A: The moduli space has been constructed: https://arxiv.org/abs/alg-geom/9410003
However, I don't think anyone proved anything more than that. In particular, I don't know if the existence of a virtual fundamental class is known for it.
A: I don't know if this has any bearing on the question, but R. Vakil has shown that moduli spaces of surfaces with very ample canonical bundle can have "arbitrarily bad" singularities.
A: I know that you are thinking firmly about the integrable world, but I thought it worth adding that for symplectic manifolds, there is no obvious generalisation of Gromov-Witten theory to higher dimensional subvarieties. This is because to define "holomorphic" you use a non-integrable almost complex structure and non-integrability means that there are no higher dimensional holomorphic objects. The fact that there are holomorphic curves can be thought of as an instance of the fact that all almost complex structures over 2-manifolds are automatically integrable. (E.g., since there are no (2,0)-forms, the space where the Nijenhuis tensor should live is zero.)
A: As ABayer said since the domain is no longer a curve, there are higher obsturction and one does not know how to define the virtual cycle. By the way, one can consider a related problem: try to count special lagrangians (in general calibrated submanifolds). It turns out that the deformation of calibrated submanifolds is unobsturcted. But one meets another problem to define a invariant, i.e the compactification issue, one usually does not know how to compactify the moduli space of calibrated submanifolds. 
