Gravity as a gauge theory Currently, (classical) gravity (General Relativity) is NOT a gauge theory (at least in the sense of a Yang-Mills theory). 
Why should "classical" gravity be some (non-trivial or "special" or extended) gauge theory? Should quantum gravity be a gauge theory?
Remark: There are some contradictory claims in the literature to this issue. To what extent gravity is "a gauge" theory?Obviously, GR is not a YM theory. So, why do some people say that gravity "IS" a gauge theory? I found this question relevant, e.g., then we handle GR in the Einstein-Cartan theory or any other theory beyond GR, like teleparallel theories or higher-derivative gravitational theories. So I think it could be useful to discuss the "gauge flavor" of gravity here.
 A: Gravity can be seen as a gauge theory of the Lorentz group (which acts on the tangent space). These was pointed out by Kibble and Sciama during the 50s and 60s.
As John  said before, it's better seen in terms of differential forms.
Another reference you might find interesting is the Lecture notes on Chern-Simons gravity by Jorge Zanelli (available in arXiv).
A: A theory is usually denominated a 'gauge theory' if all the interactions in that theory are introduced by promoting global symmetries to gauge symmetries. Note that a gauge theory is a gauge invariant theory, but a gauge invariant theory doesn't has to be a gauge theory (for example, the Standard Model is gauge invariant, but it's not a gauge theory since the scalar self-interaction doesn't enlarge the gauge symmetry of the model). Yang-Mills theory is an example of gauge theory, but not all gauge theories are of Yang-Mills type. General Relativity is a gauge theory in three different senses, namely:


*

*Invariance under diffeomorphims. Diffemorphism may be seen as a local (gauged) version of translations $\delta x^{\mu}\rightarrow a^{\mu}(x)$. In order for the theory to be diff. invariant, a covariant derivative $\nabla$ must replace partial derivatives $\partial$ (a general, dynamic metric $g$ tensor must replace Minkowski metric $\eta$ as well). Here, the most similar field to Yang-Mills connection is the Levi-Civita connection $\Gamma$ (note that in Palatini's formulation this field is independent of the metric), which transforms as a tensor plus a term that involves the derivative of $a(x)$, similar to the transformation of a non-abelian field. 

*Invariance under infinitesimal diff. One can split $g$ in a fixed background and a dynamical perturbation $h$, and the action of an infinitesimal diff. on the perturbation turns out to be $\delta h_{\mu\nu}=\partial_{\mu}a_{\nu}+\partial_{\nu}a_{\mu}$, which is also a gauge symmetry. This is the gauge symmetry connected with the masslessnes of gravitons (much like $SU_c(3)$ is related to the masslessnes of gluons and $U_{em}(1)$ to the masslenes of photons). Here, the most similar field to Yang-Mills connection is $h$, which transforms similarly to the electromagnetic potential, even though $h$ is not a connection in any sense I am aware of. 

*Invariance under local Lorentz transformations. It turns out that in order for spinors to be coupled to the gravitational field, it is convenient to introduced the tetrad formulation. In this approach, there is a gauge symmetry related to the freedom that one has to choose different basis in different space-time points. One has to introduce a covariant derivative (different from the first one in this answer) that allows us to change basis. This formulation is the closest to Yang-Mills theory (well, Ashtekar variables probably are even closer). The main difference is that in GR, besides a dynamical connection (equivalent to the gauge field in Yang-Mills), there is a tetrad field (due to the fact that the metric is a dynamical field in gravity) that doesn't have a counterpart in Yang-Mills. Here, the closest field to Yang-Mills' is the before-mentioned spin connection, which transforms as a tensor plus a term that involves the derivative of the local Lorentz transformation, very similarly to a Yang-Mills field.
A: Gravity isn't Yang-Mills theory in the narrow sense – well, except for equivalences such as AdS/CFT or Matrix theory that imply that a quantum gravitational theory is fully equivalent to a gauge theory living in a different space (e.g. in AdS/CFT, on the boundary of the AdS space).
However, gravity is a gauge theory in the broader sense because it's conveniently formulated using the diffeomorphism symmetry group. The diffeomorphisms identify physical configurations that are physically equivalent, just like in the Yang-Mills case, so although they are not of the Yang-Mills type, they have to be treated just like Yang-Mills symmetries in Yang-Mills theories.
