I will discuss next-order asymptotics as n goes to infinity, for the minimum energy configurations of n particles minimizing an inverse-power-law interaction energy. This is a question related to Random Matrix Theory, to Approximation Theory and to Statistical Physics. The first-order term in our large-n asymptotics was known since the 30's. The next-order term, obtained in collaboration with Sylvia Serfaty, is a new functional on micro-scale asymptotic configurations of the points. I will describe some more precise rigidity results (part of joint work with Simona Rota-Nodari) regarding the uniformity of such micro-scale configurations, which is a possible step towards the Abrikosov crystallization conjecture. The related study of energy-minimizing lattices (joint work with Laurent Betermin) and the quantum version of the problem (joint work with Codina Cotar) will also be mentioned.