We develop a theory of bound states in the continuum (BICs) in multipolar lattices—periodic arrays of resonant multipoles. We predict that BICs are completely robust to changes in the lattice parameters, remaining pinned to specific directions in the $k$ space. The lack of radiation for BICs in such structures is protected by the symmetry of multipoles forming the lattice. We also show that some multipolar lattices can host BICs forming a continuous line in the $k$ space, and such BICs carry zero topological charge. The developed approach sets a direct fundamental relation between the topological charge of BICs and the asymptotic behavior of the $Q$ factor in its vicinity. We believe that our theory is a significant step towards gaining deeper insight into the physics of BICs and the engineering of high-$Q$ states in all-dielectric metasurfaces.

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• Mie scattering
• Optical microcavities
• Scattering theory
• Metasurfaces
• Band structure methods