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The class of second-order curvilinear gratings consists of all the curvilinear gratings that are obtained by second-order spatial transformations of periodic gratings. It includes, for example, circular, elliptic, and hyperbolic gratings as well as circular, elliptic, and hyperbolic zone plates. Such structures occur quite frequently in optics, and their Fourier transforms may arise, for instance, in connection with the Fraunhofer diffraction patterns generated by these structures. I present the two-dimensional Fourier spectra of the most important second-order curvilinear gratings for gratings having any desired intensity profile (cosinusoidal, sawtooth wave, square wave, etc.). These analytic results are also illustrated by figures showing the various gratings and their spectra as they are obtained on a computer by two-dimensional fast Fourier transform