The complexity of turbulence has attracted for centuries the interest of scientists, philosophers and poets. Images and metaphors of turbulence of impetuous rivers and stormy seas are ubiquitous in literature, and the drawings and scripts of Leonardo da Vinci can be considered the earliest scientific studies on turbulence, which grasped some realistic details of the problem.
On the other hands, the interest for turbulence is clearly understandable because of its practical relevance in applications ranging from naval and aeronautical engineering to climate studies and weather forecast.
In the last century the works of L. Euler, L. M. H. Navier, G. G. Stokes, O. Reynolds, has given the basis of a research field which is still open nowadays. Even if the equations which rule the turbulent behavior are well known, a complete understanding of the matter is still lacking.
In this chapter I will present a short introduction to the basic concepts and phenomenology of the classical theory of turbulence, with a particular attention to the case of two-dimensional turbulence, stressing the differences and similarities with the three-dimensional case. The aim is not to provide a review of the matter which can be found in [1,2,3,4,5,6], but just to introduce, for the sake of self-consistency, the terms and concepts that will be used in the thesis.