The Pauli Exclusion Principle is a basic principle of Quantum Mechanics, and its validity has never been seriously challenged. However, given its fundamental standing, it is very important to check it as thoroughly as possible. Here we describe the VIP (VIolation of the Pauli exclusion principle) experiment, an improved version of the Ramberg and Snow experiment (E. Ramberg and G. Snow, Phys. Lett. B238, 438 (1990)); VIP has just completed the installation at the Gran Sasso underground laboratory, and aims to test the Pauli Exclusion Principle for electrons with unprecedented accuracy, down to eta^2/2 approx 10^{-30} -- 10^{-31}. We report preliminary experimental results and briefly discuss some of the implications of a possible violation.