The standard Cold Dark Matter (CDM) cosmological model provides a good description of a wide
range of astrophysical and cosmological data. However, there are a few big open questions that make the
standard model look like an approximation to a more realistic scenario yet to be found. In this paper,
we list a few important goals that need to be addressed in the next decade, taking into account the
current discordances between the different cosmological probes, such as the disagreement in the value
of the Hubble constant H0, the σ8–S8 tension, and other less statistically significant anomalies. While
these discordances can still be in part the result of systematic errors, their persistence after several years
of accurate analysis strongly hints at cracks in the standard cosmological scenario and the necessity for
new physics or generalisations beyond the standard model. In this paper, we focus on the 5.0 σ tension
between the Planck CMB estimate of the Hubble constant H0 and the SH0ES collaboration measurements.
After showing the H0 evaluations made from different teams using different methods and geometric
calibrations, we list a few interesting new physics models that could alleviate this tension and discuss
how the next decade’s experiments will be crucial. Moreover, we focus on the tension of the Planck
CMB data with weak lensing measurements and redshift surveys, about the value of the matter energy
density m, and the amplitude or rate of the growth of structure (σ8, f σ8). We list a few interesting
models proposed for alleviating this tension, and we discuss the importance of trying to fit a full array
of data with a single model and not just one parameter at a time. Additionally, we present a wide range
of other less discussed anomalies at a statistical significance level lower than the H0–S8 tensions which
may also constitute hints towards new physics, and we discuss possible generic theoretical approaches
that can collectively explain the non-standard nature of these signals. Finally, we give an overview of
upgraded experiments and next-generation space missions and facilities on Earth that will be of crucial
importance to address all these open questions.