This book describes the relation between profinite semigroups and symbolic dynamics. Profinite semigroups are topological
semigroups which are compact and residually finite. In particular, free profinite semigroups can be seen as the completion
of free semigroups with respect to the profinite metric. In this metric, two words are close if one needs a morphism on a
large finite monoid to distinguish them. The main focus is on a natural correspondence between minimal shift spaces (closed
shift-invariant sets of two-sided infinite words) and maximal J-classes (certain subsets of free profinite semigroups). This
correspondence sheds light on many aspects of both profinite semigroups and symbolic dynamics. For example, the return words
to a given word in a shift space can be related to the generators of the group of the corresponding J-class. The book is aimed
at researchers and graduate students in mathematics or theoretical computer science.