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We analyze theoretically the light transmission characteristics of corrugated long-period gratings formed in slab waveguides. The transmission spectra of the gratings show distinct rejection bands at specific wavelengths, known as the resonance wavelengths. We investigate in detail the phase-matching curves of the gratings, which govern the relationship between the resonance wavelength and the grating period. Thanks to the flexibility in the choice of the waveguide parameters, the phase-matching curves of a long-period waveguide grating can be different characteristically from those of a long-period fiber grating (LPFG), which implies that the former can exhibit much richer characteristics than the latter. Unlike an LPFG, the transmission spectrum of a long-period waveguide grating is in general sensitive to the polarization of light. Nevertheless, a proper choice of the waveguide and grating parameters can result in a polarization-independent rejection band. Long-period waveguide gratings should find potential applications in a wide range of integrated-optic waveguide devices and sensors.