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Magnetization reversal in arrays of magnetic nanoparticles with perpendicular anisotropy is investigated. Aside from domain-wall propagation effects, the reversal involves two main aspects: the nucleation behavior of individual particles and interparticle interactions. Due to magnetostatic self-interaction effects, the interparticle interaction cannot be reduced to a stray-field correction to the external field. Exact nucleation fields and explicit stray-field and self-interaction contributions are obtained for rings of equidistant dots. An exact treatment of self-interactions in various structurally inhomogeneous but rotationally symmetric wire, sphere, and thin-film nanostructures leads to renormalization of the uniaxial anisotropy. Finally, an approximate method to calculate nucleation fields is discussed.