Roughly speaking, periods are numbers that arise as integrals of polynomial differential forms over sets that are cut out by polynomial equations. More conceptually, periods are numbers that arise from the natural pairing between singular and algebraic de Rham cohomology of varieties (or more generally, motives) over $\mathbb{Q}$.

Some examples of periods are algebraic numbers, $\pi$, $\log(2)$ and other special values of logarithm, and special values of Riemann's zeta function. It is expected that every algebraic relation between periods should "come from geometry": this is the moral of Grothendieck's period conjecture, a very deep and fascinating conjecture of Grothendieck that lies in the heart of the intersection of number theory, algebra, and geometry. The goal of this talk is to give an introduction to periods and some of the fundamental conjectures about them.