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This paper is devoted to an in-depth discussion of the Gaussian Noise (GN) model which describes non-linear propagation in uncompensated coherent transmission systems. Similar models and validation efforts are reviewed. Then, the main equations of the GN model are introduced. An intuitive physical interpretation of the equations and their features is proposed. The main characteristics of the non-linear interference (NLI) noise spectra that the GN model produces are discussed. To ease model exploitation, a new formulation in hyperbolic coordinates is proposed, which makes numerical integration faster. New approximate closed-form solutions are also provided. An extension of the GN model to distributed-amplification scenarios is introduced. NLI noise accumulation versus distance and bandwidth are studied in depth. Finally, the GN model implications as to system and networks design and optimization are discussed.