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According to the standard cosmological (ΛCDM) model, the universe today is mainly composed by a cosmological constant, denoted by Λ, and by Cold Dark Matter (CDM). Whereas this paradigm is in tremendous agreement with Cosmic Microwave Background (CMB) and Large-Scale Structure (LSS) data, some discrepancies exist, e.g., on the cosmological and local determination of the Hubble parameter H0, and on the measurement of the amplitude of the matter fluctuations, σ8. Furthermore, assuming the ΛCDM model, cosmological N-body simulations exhibit some limits in reproducing the observed DM astrophysical properties at (sub-)galactic scales. The inclusion of baryon feedback is crucial to give a realistic picture, and it shows that baryons can indeed mitigate such CDM “small-scale crisis”. However, in the absence of a fully satisfactory solution within the ΛCDM framework, alternative DM scenarios emerged as a possible way to explain the tensions. The detailed small-scale features of the matter power power spectrum depend on the fundamental DM nature, allowing for a direct link between theoretical particle physics models and astrophysical observations. In fact, many non-standard DM candidates have been shown to provide a better description of structure formation at small scales, with respect to the ΛCDM model. In this talk, I will present a flexible and efficient method to constrain several classes of DM scenarios inducing small-scale departures from the ΛCDM model, against the observed properties of the Inter-Galactic Medium (IGM), which main manifestation, the Lyman-alpha forest, is an ideal tracer for the DM distribution on such scales. I will show how to translate such limits to bounds on the fundamental DM properties, without the need to run any specific numerical simulations. Finally, I will focus on a selection of viable non-standard DM candidates, such as ultra-light axions, interacting DM, and primordial black holes, and discuss the most updated limits obtained through high-resolution and high-redshift Lyman-alpha forest data alone, as well as in combination with other cosmological probes, such as Cosmic Microwave Background (CMB) and Baryon Acoustic Oscillation (BAO).