The present book contains an outline of the modern theory of iterative functional equations. The expression functional equations is here understood in a narrow sense (equations of finite kind; see Kuczma [20]). It does not include equations in which infinitesimal operations are performed on the unknown functions. So, e.g., differential equations with transformed argument do not fall under this notion.

Nowadays, mainly owing to various activities of Professor J. Aczél, functional equations have grown to be a large, independent branch of mathematics, with its own methods, rich in results and abounding in applications. In such a large area further subdivisions are indispensable. The main line of division runs between equations in several variables in which at least one unknown function depends on fewer variables than the number of independent variables actually occurring in the equation (see Aczél [2], Kuczma [12]) and equations in a single variable, which can be written using one independent variable only.

Functional equations containing several variables are dealt with in another Encyclopedia volume written by J. Aczél and J. Dhombres [1]. The reader interested in the history of functional equations can consult Dhombres [4], [2] and also Aczél [3], Aczél–Dhombres [1].

‘Iterative functional equations’ is just another name for functional equations in a single variable (such equations are also referred to as equations of rank 1). Thus the subject matter of this book is approximately the same as that of Kuczma [26].