The set of real numbers and the set of complex numbers have the power of continuum. Among these numbers, those which are ``interesting'', which appear ``naturally'', which deserve our attention, form a countable set. Starting from this point of view we are interested in the periods as defined by M.~Kontsevich and D.~Zagier. We give the state of the art on the question of the arithmetic nature of these numbers: to decide whether a period is a rational number, an irrational algebraic number or else a transcendental number is the object of a few theorems and of many conjectures. We also consider the approximation of such numbers by rational or algebraic numbers.