Unitons and their moduli

We sketch the proof that unitons (harmonic spheres in $\operatorname {U}(N)$) correspond to holomorphic `uniton bundles', and that these admit monad representations analogous to Donaldson's representation of instanton bundles. We also give a closed-form expression for the unitons involving only matrix operations, a finite-gap result (two-unitons have energy $\ge 4$), computations of fundamental groups of energy $\le 4$ components, new methods of proving discreteness of the energy spectrum and of Wood's Rationality Conjecture, a discussion of the maps into complex Grassmannians and some open problems.