Cosmological tensions in the era of precision cosmology: Insights from Tensions in Cosmology 2025

The ``Tensions in Cosmology'' series of conferences has been established as one of the main venues where the cosmological community collectively assesses the cracks in the concordance model and explores possible theoretical and observational remedies. The 2025 edition, held once again in Corfu, Greece, came at a crucial time: the Hubble constant $H_0$ discrepancy has now exceeded $6\sigma$, and new high-precision data from DESI, JWST, ACT, and other facilities have made this tension more robust while opening new windows on the early and late Universe. The $S_8$ tension, though milder and survey-dependent, remains an important probe of late-time structure formation, while emerging anomalies involving dynamical dark energy and neutrino physics are gaining increasing attention as potential signs of physics beyond $\Lambda$CDM. Here we provide a report on the meeting and an update on the state of the tensions in 2025, highlighting progress since the pioneering 2022 event.