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Kloosterman sums appear naturally in many situations of modern analytic number theory. Classically, these are defined by certain exponential sums. These sums play a central role in the study of modular forms, the distribution of primes, and the spectral theory of automorphic functions.  In this talk, we will study questions related to the distribution of Kloosterman sum angles.   It is well known that the vertical family of Kloosterman angles are equidistributed in the unit interval according to the Sato-Tate measure. In the 1990s, Katz conjectured about the spacing statistics for the "straightened" Kloosterman angles. Motivated by Katz's conjecture, we investigate problems related to the finer distribution of these angles.