The group lead by García de Abajo have explored the ability of two-dimensional periodic arrays of atoms to produce light amplification and generate laser emission when optical gain is introduced in the system by means of external illumination (i.e., an optical pump). Specifically, they have predicted that light emission of a laser type can take place for arbitrarily weak atomic scatterers (i.e., regardless of how weak is the interaction between light and the atomic transition used for the laser emission), assisted by cooperative interaction among atoms in a 2D periodic lattice. They have based this conclusion on analytical theory for three-level atoms, which additionally revealed a rich interplay between lattice and atomic resonances. Lattice resonances are well known in optics and lie at the heart of widely known phenomena such as the Wood anomalies and extraordinary optical transmission (i.e., funneling of light through arrays of small holes). Their results should provide a general background to understand light amplification and emission of laser light in periodic atomic arrays, with promising applications in the generation, manipulation, and control of coherent photon states at the nanoscale.