Fermat Quotients in 3D: Divisibility, Distribution and Dynamics

We give a survey of various arithmetic properties of Fermat Quotients $q_p(a)= (a^{p-1} -1)/p$ such as p-divisibility, distribution in residue classes modulo $p$, and properties of the dynamical system $x \mapsto q_p(x) \pmod p$. These results are related to the classical questions about Wieferich primes, yet their study requires a combination of several modern techniques coming from additive combinatorics, sieve methods, the distribution of smooth numbers and bounds of Heilbronn exponential sums.