The main problem of enumerative combinatorics is to count the elements of a finite set. Problems of this type appear in a broad spectrum of contemporary mathematics. The methods that have been developed for their solution are especially important within discrete mathematics. This chapter introduces some basic principles of enumeration and provides applications to elementary problems of enumeration and of computation of combinatorial sums. The concept of a bijection (one-to-one correspondence) of finite sets will be of primary importance.