Summary: The paper presents a systematic theory for asymptotic inferences based on autocovariances of stationary processes. We consider nonparametric tests for serial correlations using the maximum (or \(\mathcal{L}^\infty\)) and the quadratic (or \(\mathcal{L}^2\)) deviations of sample autocovariances. For these two cases, with proper centering and rescaling, the asymptotic distributions of the deviations are Gumbel and Gaussian, respectively. To establish such an asymptotic theory, as byproducts, we develop a normal comparison principle and propose a sufficient condition for summability of joint cumulants of stationary processes. We adapt a blocks of blocks bootstrapping procedure [\textit{H. R. Künsch}, Ann. Stat. 17, No. 3, 1217--1241 (1989; Zbl 0684.62035); \textit{R. Y. Liu} and \textit{K. Singh}, in: Exploring the limits of bootstrap: Papers presented at a special topics meeting, East Lansing, UK, May 1990. New York, NY: Wiley. 225--248 (1992; Zbl 0838.62036)] to the \(\mathcal{L}^\infty\) based tests to improve the finite-sample performance.