We give a survey of various arithmetic properties of Fermat Quotients qp(a) = (ap-1 - 1) / p such as p-divisibility, distribution in residue classes modulo p, and properties of the dynamical system x → qp(x) (mod 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.