  Ordered lattices of emitters with subwavelength periodicities support
unconventional forms of light-matter interactions arising from collective
effects. Here, we propose the realization and control of subradiant optical
states within the radiation continuum in two-dimensional lattices. We show how
bound states in the continuum (BICs) which are completely decoupled from
radiative states emerge in non-Bravais lattices of emitters. Symmetry breaking
results in quasi-BICs with greatly extended lifetimes, which can be exploited
for quantum information storage. The analytical derivation of a generalized
effective polarizability tensor allows us to study the optical response of
these arrays. We discuss how thanks to the quasi-BICs, a rich phenomenology
takes place in the reflectivity spectrum, with asymmetric Fano resonances and
an electromagnetically induced transparency window. Finally, we exploit these
lattices as quantum metasurfaces acting as efficient light polarizers.
