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FFT in two dimensions?

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How do I do FFT in two dimensions (for an image)? I already found free public domain (I think/hope) code for running 1-dimensional FFT. I do seem to recall someone some time long ago mentioning that you could use 1d FFT code for 2d FFT if you were clever, but I don't remember how it was done, or if it was at all correct. Or do I need special code for 2d FFT? If so, where could I find it? Btw, the existing code represents the input and output as an array of complex numbers, and a template argument specifies which complex number class is used. Only requirement is that it must provide the methods real() and imag().

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