Periods are integrals of meromorphic differential forms on compact algebraic varieties defined over rational numbers,
whose coefficients are rational numbers, along chains whose boundaries
are contained in subvarieties defined by equations with rational coefficients.

Periods form a countable set which is next in line after the (countable) set of algebraic numbers.

I will explain what do we know about periods thanks to the last 10-15 years advances in arithmetic algebraic geometry, and we do want to know about them.