  The discovery of ultracold dilute liquids has significantly elevated our
interest in various phenomena which go under the name of beyond-mean-field
(BMF) physics. In these lecture notes we give an elementary introduction to the
quantum stabilization and liquefaction of a collapsing weakly interacting
Bose-Bose mixture. A detailed derivation of the leading BMF correction, also
known as the Lee-Huang-Yang (LHY) term, in this system is presented in a manner
suitable for further generalizations and extensions. Although the LHY term is a
nonanalytic function of the density $n$, under certain conditions the leading
BMF correction becomes analytic and can be expanded in integer powers of $n$,
effectively introducing three-body and higher-order interactions. We discuss
why and how well the Bogoliubov approach can predict these few-body
observables.
