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A new approach is presented for investigating the superposition of any number of periodic structures, and the Moire effects which may result. This approach, which is based on an algebraic analysis of the Fourier-spectrum using concepts from the theory of geometry of numbers, fully explains the properties of the superposition of periodic layers and of their Moire effects. It provides the fundamental notations and tools for investigating, both in the spectral domain and in the image domain, properties of the superposition as a whole (such as periodicity or almost-periodicity), and properties of each of the individual Moire generated in the superposition (such as their profile forms and intensity levels, their singular states, etc.). This new, rather unexpected combination of Fourier theory and geometry of numbers proves very useful, and it offers a profound insight into the structure of the spectrum of the layer superposition and the corresponding properties back in the image domain