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We introduce a ``geometric'' method to bound periods of automorphic forms.The key features of this method are the use of equidistribution results inplace of mean value theorems, and the systematic use of mixing and the spectralgap. Applications are given to equidistribution of sparse subsets of horocyclesand to equidistribution of CM points; to subconvexity of the triple productperiod in the level aspect over number fields, which implies subconvexity forcertain standard and Rankin-Selberg $L$-functions; and to bounding Fouriercoefficients of automorphic forms.

Sparse equidistribution problems, period bounds and subconvexity