Hunting the 'impossible atoms' Pauli exclusion principle violation and spontaneous collapse of the wave function at test

The Pauli exclusion principle (PEP) and, more generally, the spin-statistics connection, are at the very basis of our understanding of matter, life and Universe. The PEP spurs, presently, a lively debate on its possible limits, deeply rooted in the very foundations of Quantum Mechanics. It is, therefore, extremely important to test the limits of its validity. The Violation of the PEP (VIP) experiment established the best limit on the probability that PEP is violated by electrons, using the method of searching for PEP forbidden atomic transitions in copper. We describe the experimental method, the obtained results, and plans to go beyond the actual limit by upgrading the experimental apparatus. We discuss the possibility of using a similar experimental technique to search for X-rays as a signature of the spontaneous collapse of the wave function predicted by continuous spontaneous localization (CSL) theories.