We study the energy spectrum of magnons in a ferromagnet with topologically nontrivial magnetization profile. In the case of inhomogeneous magnetization corresponding to a metastable state of ferromagnet, the spin-wave equation of motion acquires a gauge potential leading to a Berry phase for the magnons propagating along a closed contour. The effect of magnetic anisotropy is crucial for the Berry phase; we show that the anisotropy suppresses its magnitude, which makes the Berry phase observable in some cases, similar to the Aharonov-Bohm effect for electrons. For example, it can be observed in the interference of spin waves propagating in mesoscopic rings. We discuss the effect of domain walls on the interference in ferromagnetic rings, and propose some experiments with a certain geometry of magnetization. We also show that the non-vanishing average of topological field acts on the magnons like a uniform magntic field on electrons. It leads to the quantization of the magnon spectrum in the topological field.