It is well known that continuity has a primary importance for the Philosophy of Mathematics. That is why the investigation into the nature of the continuum has been the object of constant effort and trial for many scientists since antiquity. Although this concept is intuitively very clear and simple, its exact definition, without any intuitive elements, always constituted a very difficult problem. It seemed impossible to most ancient philosophers to analyse this notion into more elementary ones. Nor did others think continuity was a logical or mathematical idea, but rather a pure dogma. As a matter of fact, Greek mathematicians succeeded in approaching the geometrical conception of the continuum through an arithmetical one, by means of the invention of incommensurable numbers. This greek theory could hardly be completed in two thousand of years. What recent mathematicians did, was a representation of the geometrical continuum merely. Thus, the class of all real numbers constituted the linear continuum of Geometry. Later on, by the invention of the set-theory an attempt was made to approach the continuum through the discrete. The method inaugurated in this way has had the intention to bridge the abyss existing between these two quite different concepts, i.e. the concept of continuity and that of discontinuity (discreteness). We have already noted that most of the Greek philosophers thought it appropriate to get continuity as their starting point, inasmuch as they considered continuity and not discontinuity to be the simpler concept. In this paper, after a historical report of former debates concerning the problem of continuity, we intend to give the modern position on this problem on account of the new achievements of the set-theory and the aspects of the Intutitionistic School as well.