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Satellite communications industry is currently going through a rapid and profound transformation to adapt to the recent innovations and developments in the realm of non-geostationary orbit (NGSO) satellites. The growing popularity of NGSO systems, with cheap manufacturing and launching costs, has set to revolutionize the internet market. In this context, accurate channel characterization is crucial for the performance optimization and designing efficient NGSO communications, especially considering the dynamic propagation environment. While the Third Generation Partnership Project (3GPP) has provided some guidelines in Release 15, we observed certain divergence on the channel models considered in the literature, each with different assumptions and peculiarities. This paper provides an extensive review of the existing methods proposed for NGSO channel modeling that consider different orbits, frequency bands, user equipment, use-case and scenario peculiarities. The provided review discusses the channel modeling efforts from a contemporary perspective through trade-off analyses, classifications, and highlighting their advantages and pitfalls. The main goal is to provide a comprehensive overview of NGSO channel models to facilitate the selection of the most appropriate channel based on the scenario requirements to be evaluated and/or analysed.