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This is a review addressing soliton-like states in systems with nonlocal nonlinearity. The work on this topic has long history. Some findings, such as optical solitons supported by thermal nonlinearity, and by the orientational nonlinearity in liquid crystals, have been reviewed in the literature, therefore they are outlined in the present review in a brief form. Some other studies, such as those addressing models with fractional diffraction, which is represented by a linear nonlocal operator, have started recently, therefore it will be relevant to review them in detail when more results are accumulated; the present article provides a short outline of the latter topic. The main part of the article is a summary of results obtained for two-dimensional (2D) solitons in specific models originating in studies of Bose-Einstein condensates (BECs), which are sufficiently mature but have not yet been reviewed. These are, in particular, anisotropic quasi-2D solitons supported by long-range dipole-dipole interactions in a condensate of magnetic atoms, and giant vortex solitons, which are stable for high values of the winding number, as well as 2D vortex solitons of the latter type moving with self-acceleration. The vortex solitons are states of a hybrid type, which include matter-wave and electromagnetic-wave components. They are supported, in a binary BEC composed of two different atomic states, by the resonant interaction of the two-component matter waves with a microwave field which couples the two atomic states. Some other topics are included in the review in a brief form.