The notions of integrable boundary conditions for integrable PDEs will be first explained. This can be simply realized as certain extra compatibility conditions on top of the Lax pair. Then, the inverse scattering transform and the associated Riemann-Hilbert problem can be constructed for integrable PDEs equipped with integrable boundary conditions. We will show formalism for the nonlinear Schr\{o}dinger equations. We will also extend this notion to fully discrete integrable quad-graph systems, and to present certain half-plane graph problems with possible connections to discrete complex analysis.

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