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Spin-wave modes in magnetic transition-metal nanowires having diameters of about 10 nm and lengths on the order of 1 µm are investigated by model calculations. There are quasicontinuous modes with k vectors parallel to the wires axis and discrete modes with k vectors perpendicular to the wire axis. Due to the small cross section of the wires, the perpendicular modes can be ignored in many cases and the low-temperature behavior of the wires is quasi one-dimensional. Using an analytic approach and exploiting the analogy between micromagnetism and quantum mechanics it is shown that all spin-wave modes with k vectors parallel to the wire axis are localized. The spin-wave localization is a micromagnetic analog to the Anderson localization of conduction electrons due to randomness in less than two dimensions, and, as in the electron analogy, arbitrarily small disorder is sufficient to cause the localization of all modes.