It the first part of talk we will mention the definition and general informations on the Gelfand-Kirillov dimension, and its connections with noncommutative ring theory, noncommutative projective algebraic geometry and Lie Algebras. We will mention open questions and results on the Gelfand-Kirillov dimension of domains, primitive algebras, algebraic algebras and Golod-Shafarevich algebras, and modules and rings satysfying some special relations.