  We introduce a ``geometric'' method to bound periods of automorphic forms.
The key features of this method are the use of equidistribution results in
place of mean value theorems, and the systematic use of mixing and the spectral
gap. Applications are given to equidistribution of sparse subsets of horocycles
and to equidistribution of CM points; to subconvexity of the triple product
period in the level aspect over number fields, which implies subconvexity for
certain standard and Rankin-Selberg $L$-functions; and to bounding Fourier
coefficients of automorphic forms.
