  In the algebraic theory of codes and formal languages, the set $Q$ of all
primitive words over some alphabet $\zi $ has received special interest. With
this survey article we give an overview about relevant research to this topic
during the last twenty years including own investigations and some new results.
In Section 1 after recalling the most important notions from formal language
theory we illustrate the connection between coding theory and primitive words
by some facts. We define primitive words as words having only a trivial
representation as the power of another word. Nonprimitive words (without the
empty word) are exactly the periodic words. Every nonempty word is a power of
an uniquely determined primitive word which is called the root of the former
one. The set of all roots of nonempty words of a language is called the root of
the language. The primitive words have interesting combinatorial properties
which we consider in Section 2. In Section 3 we investigate the relationship
between the set $Q$ of all primitive words over some fixed alphabet and the
language classes of the Chomsky Hierarchy and the contextual languages over the
same alphabet. The computational complexity of the set $Q$ and of the roots of
languages are considered in Section 4. The set of all powers of the same degree
of all words from a language is the power of this language. We examine the
powers of languages for different sets of exponents, and especially their
regularity and context-freeness, in Section 5, and the decidability of
appropriate questions in Section 6. Section 7 is dedicated to several
generalizations of the notions of periodicity and primitivity of words.
