This paper concerns a class of complex numbers, called periods, that appear naturally when comparing two cohomology
theories for algebraic varieties (the first defined topologically and the second algebraically). Our goal is to explain the fundamental conjectures of Grothendieck and Kontsevich–Zagier that give very precise information about the transcendence properties of periods. The notion of motive (due to Grothendieck) plays an important conceptual role. Finally, we explain a geometric version of these conjectures. In contrast with the original conjectures whose solution seems to lie in a very distant future, if at all it exists, a solution for the geometric conjectures is within reach of the actual motivic technology.