Impurity physics has emerged as a new branch of research in the field of atomic quantum gases. A central feature is the wide tunability of interactions between the impurities and the surrounding medium. By using magnetically controlled Feshbach resonances, regimes of strong interactions can be reached, which reveal intriguing many-body physics. After a general introduction into the field, I will present our experiments on fermionic and bosonic potassium impurities immersed in a deeply degenerate Fermi sea of lithium atoms. For fermionic impurities, we study the spectrum of quasiparticle excitations and the regime where the Fermi-liquid picture breaks down. Moreover, we observe the formation dynamics of quasiparticles in real time. For bosonic impurities, we observe small-sized Bose-Einstein condensates and, for strong repulsive interactions, their phase separation from the Fermi sea. If time permits, I will also introduce a new quantum gas mixture (dysprosium and potassium) with great prospects for future research on fermionic quantum gases.