Probabilistic Seismic Hazard Assessment (PSHA) plays a fundamental role in the defence strategy from earthquake destruction of most countries. In light of this paramount importancePSHA should be rooted on solid ground but it is not, since it ignores earthquake Physics starting from its most important phenomenological features. These are scale invariance, which allows to infer the behaviour of large earthquakes from smaller seismicity for which copious data are available, and clustering in time and space, which states that the more earthquakes one sees the more he should expect. The latter is just the opposite of the Characteristic Earthquake, ubiquitously used in Seismology and Geology under the paradigm of elastic rebound. As a consequence, PSHA estimates are essentially speculative and void of scientific significance. In practical terms, while PSHA has the merit of raising attention on an important problem, its faulty physical and statistical premises lead it to untrustworthy results. A substantial improvement comes from appropriately basing estimates on earthquake physical phenomenology, and realistically evaluating and reporting all uncertainties.