Cosmologists are puzzled by tensions between the results of several categories of observations, which have been growing over the past few years. The “Hubble tension” arises from different model-dependent estimates of the current expansion rate of the universe, and the “S8 tension” from different model-dependent reconstructions of the growth of matter fluctuations in the recent universe. Leaving the investigation of systematics to experts of such observations, we will explore the point of view that these anomalies could be hints of new physical ingredients in the cosmological model. I will give examples of the kind of ingredients that solve the tensions, with a particular focus on non-minimal models for dark matter, neutrinos or other relic particles in the universe. We will see that the price to pay is high: until now, only very specific and non-trivial models offer possible solutions. On the one hand, these models may seem very ad hoc, but on the other hand they can make testable predictions.