Abstract: We explain how to use direct limits of Hilbert spaces, and in particular the universal property which characterizes them, to construct wavelet bases in a variety of concrete Hilbert spaces of functions. This applies to the classical situation involving dilation matrices on L2(Rn), the wavelets on fractals studied by Dutkay and Jorgensen, and Hilbert spaces of functions on solenoids. This is joint work with Baggett, Packer, Raeburn and Ramsay.