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A growing number of technologies (communications, imaging, solar energy, etc) rely on manipulating linear waves at the wavelengthÌýscale, and accurate numerical modeling is key for device design.Ìý IÌýwill focus on diffraction problems where time-harmonic scalar wavesÌýscatter from piecewise-uniform periodic media.Ìý After reviewing theÌýintegral equation method, I explain two innovations that allow ourÌýsolvers to be high-order, robust, and optimal O(N) complexity: 1) newÌýsurface quadrature schemes in 2D and 3D compatible with the fastÌýmultipole method, and 2) a simple way to "periodize" the integralÌýequation so that the unknowns live on only a single period of theÌýgeometry.Ìý The resulting linear systems are solved iteratively or in aÌýfast direct fashion.Ìý I will present efficient solvers for gratingsÌýwith up to a thousand periodic interfaces or inclusions.Ìý We areÌýextending the techniques to other boundary value problems such asÌýStokes flow. ÌýThis is joint work with: Min Hyung Cho, Jun Lai, Adrianna Gillman,ÌýLeslie Greengard, Andreas Kloeckner, Mike O'Neil, Zydrunas Gimbutas.