We describe resent results (obtained in a joint work with J. Bourgain, K. Ford and S. Konyagin) about the smallest integer $a > 1$, for which the Fermat quotient $q_p(a) = (a^{p-1}-1)/p$ does not vanish modulo a prime $p$, which in improve a result of H.W. Lenstra of 1979 from $4(\log p)2$ down to $(\log p)^{463/252 + o(1)}$, for all $p$, and down to $(\log p)^{5/3 + o(1)}$, for almost all $p$. We also discuss recent results (obtained in a joint work with A. Ostafe) about some dynamical properties of the map $a\mapsto q_p(a) (mod p)$ such as the cycles length and the number of fixed points. Underlying techniques include results on the distribution of smooth numbers and elements of multiplicative subgroups of residue rings, bounds of Heilbronn exponential sums and a large sieve inequality with square moduli will be discussed too.