  The standard $\Lambda$ Cold Dark Matter cosmological model provides an
amazing description of a wide range of astrophysical and astronomical data.
However, there are a few big open questions, that make the standard model look
like a first-order approximation to a more realistic scenario that still needs
to be fully understood. In this Letter of Interest we will list a few important
goals that need to be addressed in the next decade, also taking into account
the current discordances present between the different cosmological probes, as
the Hubble constant $H_0$ value, the $\sigma_8 - S_8$ tension, and the
anomalies present in the Planck results. Finally, we will 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 questions.
