Gluon scattering in N=4 super-Yang-Mills theory from weak to strong coupling

I describe some recent developments in the understanding of gluon scattering amplitudes in N=4 super-Yang-Mills theory in the large-N_c limit. These amplitudes can be computed to high orders in the weak coupling expansion, and also now at strong coupling using the AdS/CFT correspondence. They hold the promise of being solvable to all orders in the gauge coupling, with the help of techniques based on integrability. They are intimately related to expectation values for polygonal Wilson loops composed of light-like segments.