The real Turing machine is a computational model that has been proposed by Blum, Shub and Smale to model computations on real numbers. This model allows to understand complexity of problems over the real numbers in terms of the number of algebraic operations required to solve them, independently from machine representations. Complexity classes like P, NP, ... with complete problems, can be defined in this model. We will present several results that show that complexity classes can be characterized algebraically in this model. These results, which are true on any logical structure, extend known characterizations in classical complexity, in the spirit of the first characterization of polynomial time obtained in 1992 by Bellantoni and Cook. Joint Work with F. Cucker, P. Jacobé de Naurois, J. Y. Marion (PhD Thesis of P. Jacobé de Naurois).