excitons [7], because there is no charge transfer between atoms and the dipole interaction is due to bound electrons. Some of the locsiton effects, first of all, wave resonances, may be viewed as analogues of other types of oscillations and waves in condensed matter, like plasmons and phonons [7], as well as low-dimensional effects, like surface plasmons [8, 9], size-related resonances in thin metal films [10] or long organic molecules [11], “quantum carpets” [12], arrays of pendulums or electronic circuits [2], etc. Approaching from another perspective, one can view the formation of a locsiton band as a Rabi broadening of the atomic resonance due to strong interatomic interactions, which is essentially similar to the band formation in solid-state theory. However, although all waves and oscillations may be said to have something in common, locsitons form a distinct new class of phenomena be- cause of the nonconductive, dielectric, nature of their optical response and a strong coupling between atoms, which is needed for attaining dramatic size-related resonances, magic con- figurations, and other interesting effects. A separate issue outside of the scope of this paper is how the structures can be fabricated or arranged. One can envision placing the atoms in a controlled way on the surface of suitable dielectric materials; recent developments in the atomic- and ion-traps technology allow for arranging atoms in vacuum as 1D arrays and 2D “crystals” in the so called wire traps [13]. Our paper is structured as follows. In Sec. II we outline our problem and present some general formulas; for more details the reader should refer to our recent paper [2]. In Sec. III we describe locsitons in infinite, unbounded, 2D lattices of resonant atoms in the case when the incident field is polarized in the lattice plane. The equations for the LF and the dispersion relations for the locsitons are first obtained in the NRA and then in the more precise NNA. In Sec. IV we discuss effects that arise in finite 2D lattices, in particular, formation of 2D patterns of the LF, due to size-related locsiton resonances, and the “magic” cancellation of the resonant LF suppression. In Sec. V we describe 2D locsitons in the case when the incident field is normal to the lattice plane. Sec. VI summarizes main results of our paper. Let us consider a 2D lattice of strongly resonant identical particles, which we will further call “atoms”. We will assume that, for the incident laser frequency ω near their resonant frequency ω$_{0}$ , these atoms can be described by a two-level model with the transition dipole II. 2D LATTICES: GENERAL MODEL 5