Numbers in a given set with (or without) a large prime factor

This survey treats two connected questions in analytic number theory: given a set of natural numbers, one may seek numbers with large prime factors in the set. Alternatively, one searches for smooth numbers in the set. Many examples have been studied: the set of values of a polynomial, the set of integers in a short interval, the set of shifted primes p+a and so on. These are discussed at some length, with references to the literature.