This introductory chapter contains the basic notions: actions of groups, orbits, stabilizers, fixed points, and so on. Various examples are given. The Cauchy-Frobenius Lemma is derived, which yields the number of orbits in the case when both the group and the set on which it acts are finite.
In order to prepare the applications of this lemma and its refinements which follow in the next chapters, a detailed description of the conjugacy classes of symmetric and of monomial groups is added.
The
paradigmatic actions
which
we discuss and apply in full detail here and later on are several natural
actions on the set 
,
consisting of all the mappings from 
into 
.
These actions on are induced in a natural way
by actions of groups on or on .
The corresponding orbits are called symmetry classes of
mappings and there are many structures in mathematics and sciences which
can be defined as symmetry classes of this kind.
The enumeration of symmetry classes of mappings is described in full detail, in order to prepare refinements which are given in the following chapters.
A very simple case of a group action leads to another important enumerative concept, the so-called involution principle. Finally we discuss the enumeration of symmetry classes which consist of injective or surjective mappings only.