The Lee-Yang circle theorem is the prototype of a class of results giving very precise information on the locus of the zeros of certain families of (arbitrarily high degree)polynomials in one variable. The standard application of these results is to statistical mechanics, but we shall see that there are also results about graph-counting polynomials. The intriguing feature of all these results is that they appear quite inaccessible by more standard methods of study of polynomials.