Engle and Russell's autoregressive conditional duration (ACD) models have been widely used to model financial data that arrive at irregular intervals. In the modelling of such data, testing for duration clustering and evaluation procedures of a particular model are important steps. The portmanteau test statistics of Box and Pierce (BP) (1970) or Ljung and Box (LB) (1978) have been used by many authors. When testing for ACD effects, the test statistics are based on the raw durations. On the other side, for the adequacy of ACD models, the BP/LB procedures must be applied to the estimated standardized residuals. In this paper we propose two classes of test statistics for duration clustering and one class of test statistics for the adequacy of ACD models, using a spectral approach. The tests for ACD effects of the first class are obtained by comparing a kernel-based normalized spectral density estimator and the normalized spectral density under the null hypothesis of no ACD effects, using a norm. One member of the class provides a generalized version of the BP test statistic, using the truncated uniform kernel and the L₂ norm. However, many kernels give a higher power than the BP/LB test statistics or the truncated uniform kernel based test statistic. The second class of test statistics for ACD effects exploits the one-sided nature of the alternative hypothesis; they are based on a weighted sum of sample autocorrelations of durations, with the weighting function typically giving more (less) weight to lower (higher) orders of lags. The asymptotic distributions of the test statistics in the two classes are N(0,1) under the null hypothesis of no ACD effects. Asymptotic arguments suggest that the tests in the first class should be more powerful than the tests in the second class asymptotically, but to exploit the one-sided nature of the alternative hypothesis may be powerful in small samples. The class of tests for the adequacy of an ACD model is obtained by comparing a kernel-based spectral density estimator of the estimated standardized residuals and the null hypothesis of adequacy using a norm. The resulting test statistics possess a convenient asymptotic normal istribution under the null hypothesis of adequacy. With the L₂ norm and the truncated uniform kernel, we retrieve a generalized BP test statistic applied to the estimated standardized residuals. However, using a kernel different from the truncated uniform kernel, we obtain more powerful test procedures in many situations. We present a simulation study illustrating the merits of the proposed procedures and an application with financial data is onducted.