Wavelets are new families of basis functions that yield the representation $f(x) = \sum {b_{jk} W(2^j x - k)} $. Their construction begins with the solution $\phi (x)$ to a dilation equation with coefficients $c_k $. Then W comes from $\phi $, and the basis comes by translation and dilation of W. It is shown in Part 1 how conditions on the $c_k $ lead to approximation properties and orthogonality properties of the wavelets. Part 2 describes the recursive algorithms (also based on the $c_k $) that decompose and reconstruct f. The object of wavelets is to localize as far as possible in both time and frequency, with efficient algorithms