I will briefly review a conjectural picture of the hyperkahler metrics on moduli spaces of Higgs bundles which arose from quantum field theory, and then describe some recent progress toward proving these conjectures; this is work of various people including Mazzeo-Swoboda-Weiss-Witt, Fredrickson, Dumas-N, Fredrickson-N. A recent surprise is that for Higgs bundles of rank > 2, one needs to use “higher symmetries” which generalize the covariance of Higgs bundles under holomorphic vector fields.