The Lee-Yang circle theorem describes complex polynomials of degree n in z with all their zeros on the unit circle |z|=1. These polynomials are obtained by taking z1=⋯=zn=z in certain multiaffine polynomials Ψ(z1,…,zn) which we call Lee-Yang polynomials (they do not vanish when |z1|,…,|zn|<1 or |z1|,…,|zn|>1). We characterize the Lee-Yang polynomials Ψ in n+1 variables in terms of polynomials Φ in n variables (those such that Φ(z1,…,zn)≠0 when |z1|,…,|zn|<1). This characterization gives us a good understanding of Lee-Yang polynomials and allows us to exhibit some new examples. In the physical situation where the Ψ are temperature dependent partition functions, we find that those Ψ which are Lee-Yang polynomials for all temperatures are precisely the polynomials with pair interactions originally considered by Lee and Yang.