The ical Dirichlet $L$-series can be generalized as the multiple sums $Li\Bigl ( {z_1\ldots z_r\atop s_1\ldots s_r}\Bigr )= \sum _{(n)} {z^{n_1}_1\cdots z^{n_r}_r\over n^{s_1}_1\ldots n^{s_r}_r}$, where the summation is over systems of integers $n_1>\ldots >n_r>0$. The main question is about the arithmetical nature of the so-called colored polyzeta numbers $Li\Bigl ({\sigma _1\ldots \sigma _r\atop s_1\ldots s_r}\Bigr )$ where the $\sigma _i$ are roots of unity and the $s_i$ are integers. After a description of the ical results (Euler, Dirichlet,...), we formulate the main conjectures (Ihara, Drinfeld, Deligne, Zagier,...). We describe the state of the art in this domain, insisting on the important role played by torsors over pro-unipotent groups. We mention the positive results obtained so far by Goncharov, Écalle, Racinet, Rivoal which settle the algebraic structure of the ring $Z$ of polyzeta numbers.