*In which we give an explicit construction of expander graphs of polylogarithmic degree, state the properties of the **zig-zag product* of graphs, and provide an explicit construction of a family of constant-degree expanders using the zig-zag product and the polylogarithmic-degree construction.

A *family of expanders* is a family of graphs , , such that each graph is -regular, and the edge-expansion of each graph is at least , for an absolute constant independent of . Ideally, we would like to have such a construction for each , although it is usually enough for most applications that, for some constant and every , there is an for which the construction applies in the interval , or even the interval . We would also like the degree to be slowly growing in and, ideally, to be bounded above by an explicit constant. Today we will see a simple construction in which and a more complicated one in which .