What is the current state of the mathematics of Higgs fields? Topical. I know there are good mathematical theories in which "Higgs" is used, in a geometrical sense. Would someone care to explain?
To clarify, I'd like to know about Higgs bundles on Riemann surfaces, rather than the standard model of high-energy physics.
 A: Since you are  asking about Higgs bundles, I can say a few words here. 
These  were introduced by Hitchin in the mid 1980's, although I'm not sure he used this term.  One can look at the introduction to his paper "The self-duality equations on Riemann surfaces" for some of the motivation and background. In brief outline, he starts from a gauge theoretic perspective: the initial data is a pair consisting of a vector bundle with connection on a compact Riemann surface $X$ and additional vector valued $1$-form (the Higgs field) subject to the equations referred to above. One of his main results is that such a pair is equivalent to a pair $(E,\theta)$ -- now called a Higgs bundle -- consisting of a holomorphic vector bundle and holomorphic section $\theta\in \Gamma(\Omega_X^1\otimes End(E))$
satisfying a suitable stability condition.  One consequence is: 

Theorem. There is a correspondence between irreducible representations of  $\pi_1(X)$ and stable Higgs bundles of degree $0$. 

Simpson rightly christened (a generalization of) this the nonabelian
Hodge theorem. It is indeed a  beautiful and fundamental theorem. It  generalizes an earlier theorem of Narasimhan and Seshadri that corresponds to the case where the Higgs field is zero.  These objects also make their appearance in Geometric Langlands -- but I'm hardly the right person to discuss these aspects. I could go on, but I think I'll stop here.
Perhaps I may add that the underlying geometry is very rich. Both the representation and Higgs sides admit natural moduli problems. Under this correspondence,
the two moduli spaces are homemorphic but are analytically distinct. In fact, the complex structures behave like $i$ and $j$ of the quaternions, and together with a natural symplectic structure, constitute a hyper Kähler manifold (away from the singularities).
Such things are quite rare. On the Higgs side, there is some additional geometry:
A $\mathbb{C}^*$-action obtained by dilating the Higgs field, and the Hitchin fibration
obtained by sending a Higgs bundle $(E,\theta)$ to the characteristic polynomial of $\theta$. The latter turns out to form a completely integrable system, that is the smooth fibres are Langrangian with respect to the symplectic structure.  Arguments from symplectic geometry show that these are tori, although in fact, these fibres turn out to be abelian varieties. What is even more remarkable is that part  of this story carries over into positive characteristic and had played an important role in Ngo's work. I understand very little of this, so instead I'll direct interested  readers to  the recent Bulletin article by Nadler.
