Rare decay searches at CDF

In the last decade the CDF experiment at the Tevatron clearly demonstrated that it is possible to study extensively heavy flavour physics in hadron collisions and achieve remarkable results, competitive and complementary to $B$-factories. In this paper we report on the indirect searches for physics beyond the standard model via measurements of rare $b$-hadron decays. The final limits, based on the analysis of the full CDF data set, on the branching fraction of the $B^0_{(s)}$ decay into a pair of muons are presented and discussed. Moreover we review the latest measurements, with 6.8 fb$^{-1}$ of collected data, of the total and differential branching fractions and angular observables of rare $b$-hadron decays proceeding via the flavour-changing neutral-current process $b \rightarrow s \mu^+ \mu^-$. PACS numbers: 13.20.He, 13.30.-a, 12.15.Mn