Ultralight scalar dark matter represents the idea that the dark matter in the universe is made of boson particles with a very tiny mass, such that wave effects are present at galactic and possibly at larger scales, that could be detected by using different astrophysical and cosmological observations. The model has been studied intensely for the last two decades, starting with its behavior at cosmological scales, down to the formation of galaxies and other related cosmic structure. In this talk, I will revisit the main hypotheses of the model and its corresponding characteristic predictions, which makes it one of the most studied and constrained dark matter models in the cosmology literature. I will discuss in particular the cases of the so-called fuzzy dark matter, whose only free parameter is the boson mass, and the axion-like one that also considers the effective energy scale of the model. The topics of the talk include the cosmological dynamics, evolution of density perturbations, formation of galaxies, signals in gravitational lensing, and others. I will close with some general comments about the future perspectives of the model.