In 1844, Joseph Liouville discovered transcendental numbers — those numbers that are not roots of polynomials with rational coefficients. Nowadays, we see Liouville’s discovery as the foundation of Diophantine approximation, a fascinating subarea of number theory, which studies how well algebraic numbers can be approximated by the rationals. In my talk, I will prove the classical result of Liouville and explain further advances in the area, such as Thue’s theorem and the celebrated theorem of Roth, which enabled its discoverer to receive the Fields medal in 1958.