The extremogram is an asymptotic correlogram for extreme events constructed from a regularly varying strictly stationary sequence. Correspondingly, the spectral density generated from the extremogram is introduced as a frequency domain analog of the extremogram. Its empirical estimator is the extremal periodogram. The extremal periodogram shares numerous asymptotic properties with the periodogram of a linear process in classical time series analysis: the asymptotic distribution of the periodogram ordinates at the Fourier frequencies have a similar form and smoothed versions of the periodogram are consistent estimators of the spectral density. By proving a functional central limit theorem, the integrated extremal
periodogram can be used for constructing asymptotic tests for the hypothesis that the data come from a strictly stationary sequence with a given extremogram or extremal spectral density. A numerical method, the stationary bootstrap, can be applied to the estimation of the integrated extremal periodogram.