A Steiner quadruple system of order n is said to be cyclic if it has an n-cycle as an automorphism. Infinite classes of Steiner quadruple systems are established by exploiting the structure of PGL(2, q) (the projective linear group) for various prime powers q. Other recent results are surveyed and the known spectrum of cyclic Steiner quadruple systems for small orders is established. Several new systems are included in this discussion.