Automata-theoretic tools have been deeply utilized in literature for studying algebraic matters, in particular free monoids and free groups. In this book we propose to study finitely generated submonoids of free monoids and finitely generated subgroups of free groups, and their intersection. Free monoids play an important role in combinatorics on words and in formal language theory. The study of submonoids of free monoids has been deepened by using combinatorial and automata methods in the setting of the theory of variable-length codes, started by M.P.Schützenberger. We investigate the intersection of two finitely generated submonoids of the free monoid on a finite alphabet by using an automata-theoretic approach. For what concerns free groups, several open problems have been solved and moreover several algorithms concerning group’s problems have been optimized using automata. We focus on algorithms constructing particular bases for a subgroup of a free group and in particular strongly reduced Nielsen bases. Using inverse automata we furnish two algorithms for the construction of a strongly Nielsen basis for a finitely generated subgroup.