A class of goodness-of-fit tests based on the empirical characteristic function is studied. They can be applied to continuous as well as to discrete or mixed data with any arbitrary fixed dimension. The tests are consistent against any fixed alternative for suitable choices of the weight function involved in the definition of the test statistic. The bootstrap can be employed to estimate consistently the null distribution of the test statistic. The goodness of the bootstrap approximation and the power of some tests in this class for finite sample sizes are investigated by simulation. © 2009 Elsevier B.V. All rights reserved.