Let N(v) be the number of nonisomorphic Steiner quadruple systems. The following asymptotic estimate of N(v) is proved for all admissible orders v. \(\lim \inf_{v\to \infty}(\log N(v)/v^ 3)>0\). This result is a considerable improvement on the previous best lower bound for N(v) due to the reviewer [Congr. Numerantium 33, 45-54 (1981; Zbl 0515.05012)]. It is also close to best possible since \(\limsup_{v\to \infty}(\log N(v)/v^ 3\log v)<\infty\). The proof uses ingenious variations of several known constructions for Steiner quadruple systems. The exposition is clear and a joy to read.