Important insights into the formation and evolution of the Galactic disc(s) are contained in the chemical compositions of stars. We analysed high-resolution and high signal to noise HARPS spectra of 79 solar twin stars in order to obtain precise determinations of their atmospheric parameters, ages (σ∼0.4 Gyr) and chemical abundances (σ<0.01 dex) of 12 neutron-capture elements (Sr, Y, Zr, Ba, La, Ce, Pr, Nd, Sm, Eu, Gd, and Dy). This valuable dataset allows us to study the [X/Fe]-age relations over a time interval of ∼10 Gyr and among stars belonging to the thin and thick discs. These relations show that i) the s-process has been the main channel of nucleosynthesis of n-capture elements during the evolution of the thin disc; ii) the thick disc is rich in r-process elements which suggests that its formation has been rapid and intensive. In addition, the heavy (Ba, La, Ce) and light (Sr, Y, Zr) s-process elements revealed details on the dependence between the yields of AGB stars and the stellar mass or metallicity. Finally, we confirmed that both [Y/Mg] and [Y/Al] ratios can be employed as stellar clocks, allowing ages of solar twin stars to be estimated with an average precision of ∼0.5 Gyr.