This paper introduces new rank-based statistics for testing against serial dependence in a univariate time series context. These Kolmogorov-Smirnov and Cramér-von Mises type statistics are based on the serial version of a process originally introduced by Genest & Rivest (1993) and later studied by Barbe, Genest, Ghoudi & Rémillard (1996). Using results of Ghoudi & Rémillard (1998), the asymptotic behaviour of this serialized Kendall process and the limiting distribution of the proposed statistics are determined under the null hypothesis of randomness. Simulations comparing the power and size of the new tests and some of their parametric and nonparametric competitors are also presented. The Cramér-von Mises statistic is seen to be quite powerful against a wide range of alternatives, even for small series. Though less promising on the whole, the Kolmogorov-Smirnov statistic also does very well against certain specific types of alternatives.