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When a quantum system adiabatically traverses a closed cycle in the parameter space, the wave function of the system acquires a complex factor depending only on the cycle. This phenomenon is known as the Berry or geometric phase [1]. The Berry phase has been addressed in hundreds of studies, whose results were summarized in [2–5]. A similar effect occurs in optics, wave dynamics, and mechanics [4, 6, 7]. The Berry phase is closely related to energy level degeneracies: in the neighborhood of degeneracies, the phase can be finite for an arbitrarily small cycle in the parameter space. In this paper, general asymptotic expressions are found for the Berry phase in the neighborhood of dou-ble degeneracies. We consider both the classical case of stationary degeneracies and the case of quasi-stationary (metastable) degeneracies. In conclusion, we discuss applications to optical and mechanical systems. The results presented are based on the bifurcations theory of multiple eigenvalues [8, 9] and on the theory of versal deformations [10] and correspond to V.I. Arnold’s sug-gestion that a theory of residues be developed for calcu-lating the Berry phase [11]