Already Euler computed the values $\zeta(2), \zeta(4), \zeta(6),\ldots$ of the Riemann zeta function $\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^{s}}$ to be \begin{equation*} \zeta(2k)=-\frac{(2\pi i)^{2k}}{2(2k!)}B_{2k} \end{equation*} where $B_{2k}\in \mathbb{Q}$ are the Bernoulli numbers. This formula can be seen as the easiest case of a vast conjecture by Deligne from 1977, which relates special values of $L$-functions of arithmetic varieties and their periods. In this talk we want to give a non-technical introduction to the Deligne conjecture, aimed at general mathematical audience. In the end we discuss very recent developments, which lead to a complete proof in the case of Hecke $L$-functions.