  We review the connection of $Y$- and $Q$-systems with the BPS spectra of $4D$
$\mathcal{N}=2$ supersymmetric QFTs. For each finite BPS chamber of a
$\mathcal{N}=2$ model which is UV superconformal, one gets a periodic
$Y$-system, while for each finite BPS chamber of an asymptotically-free
$\mathcal{N}=2$ QFT one gets a $Q$-system i.e. a rational recursion all whose
solutions satisfy a linear recursion with constant coefficients (depending on
the initial conditions). For instance, the classical $ADE$ $Y$-systems of
Zamolodchikov correspond to the $ADE$ Argyres-Douglas $\mathcal{N}=2$ SCFTs,
while the usual $ADE$ $Q$-systems to pure $\mathcal{N}=2$ SYM. After having
motivated the correspondence both from the QFT and the TBA sides, and having
introduced the basic tricks of the trade, we exploit the connection to
construct and SOLVE new $Y$- and $Q$-systems.
  In particular, we present the new $Y$-systems associated to the $E_6,E_7,E_8$
Minahan-Nemeshanski SCFTs and to the $D_2(G)$ SCFTs. We also present new
$Q$-system corresponding to SYM coupled to specific matter systems such that
the YM $\beta$-function remains negative.
