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In the classical setting neighborhood structures are the standard semantic tool used for non-normal modal logics. In this paper we introduce a new fuzzy version of this semantics and we prove that it is a natural generalization in the sense that fuzzy Kripke models %(see \cite{BoEsGoRo11})can be identified as a special kind of our proposed models.In addition, we characterize this notion in a syntactic way and describe conditions on fuzzy neighborhood models sufficient for the validation of different and well known schemes.%This approach is an adaptation to the fuzzy case of the well-known%neighborhood semantics and its classical systems %\cite[Chap. 7]{Chellas80}.Finally,%we connect the notion of fuzzy neighborhood model with that of many-valued Kripke semantic.we show how this approach can be used to provide an axiomatization of a many-valued modal system capturing possibilistic reasoning over {\L}ukasiewicz logic propositions.