Investigating mixtures of bosons and fermions is an extremely active area of research in experimental physics for constructing and understanding novel quantum bound states such as those in superconductors, superfluids, and supersolids. These ultra-cold Bose-Fermi mixtures are intrinisically different from gases with only bosons or fermions. Namely, they show a fundamental instability due to energetic considerations coming from the Pauli exclusion principle. Inspired by this activity in the physics community, recently we started exploring the mathematical theory of Bose-Fermi mixtures. One of the main challenges is understanding the physical scales of the system that allow for suitable analysis. We will describe how we overcame this challenge in the joint work with Esteban Cárdenas and Joseph Miller by identifying a novel scaling regime in which the fermion distribution behaves semi-clasically, but the boson field remains quantum-mechanical. In this regime, the bosons are much lighter and more numerous than the fermions. Time permitting, we will also describe new results obtained with Esteban Cárdenas, Joseph Miller and David Mitrouskas inspired by recent experiments by DeSalvo et al. on mixtures of light fermionic atoms and heavy bosonic atoms. A key observation - and this has been theoretically long predicted - is the emergence of an attractive fermion-mediated interaction between the bosons. We give a rigorous derivation of fermion-mediated interactions and prove the associated stability-instability transition.