Optical bound states in the continuum (BICs) provide a way to engineer very narrow resonances in photonic crystals. The extended interaction time in such systems is particularly promising for enhancement of nonlinear optical processes and development of the next generation of active optical devices. However, the achievable interaction strength is limited by the purely photonic character of optical BICs. Here, we mix optical BIC in a photonic crystal slab with excitons in atomically thin semiconductor MoSe$_2$ to form nonlinear exciton-polaritons with a Rabi splitting of 27~meV, exhibiting large interaction-induced spectral blueshifts. The asymptotic BIC-like suppression of polariton radiation into far-field towards the BIC wavevector, in combination with effective reduction of excitonic disorder through motional narrowing, results in small polariton linewidths below 3~meV. Together with strongly wavevector-dependent Q-factor, this provides for enhancement and control of polariton--polariton interactions and resulting nonlinear optical effects, paving the way towards tunable BIC-based polaritonic devices for sensing, lasing, and nonlinear optics.
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