  This is a review of recent developments in the theory of beta ensembles of
random matrices and their relations with conformal filed theory (CFT). There
are (almost) no new results here. This article can serve as a guide on
appearances and studies of quantum Painlev\'e and more general multidimensional
linear equations of Belavin-Polyakov-Zamolodchikov (BPZ) type in literature. We
demonstrate how BPZ equations of CFT arise from $\beta$-ensemble eigenvalue
integrals. Quantum Painlev\'e equations are relatively simple instances of BPZ
or confluent BPZ equations, they are PDEs in two independent variables ("time"
and "space"). While CFT is known as quantum integrable theory, here we focus on
the appearing links of $\beta$-ensembles and CFT with {\it classical}
integrable structure and isomonodromy systems. The central point is to show on
the example of quantum Painlev\'e II (QPII)~\cite{betaFP1} how classical
integrable structure can be extended to general values of $\beta$ (or CFT
central charge $c$), beyond the special cases $\beta=2$ ($c=1$) and
$c\to\infty$ where its appearance is well-established. We also discuss an \'a
priori very different important approach, the ODE/IM correspondence giving
information about complex quantum integrable models, e.g.~CFT, from some
stationary Schr\"odinger ODEs. Solution of the ODEs depends on (discrete)
symmetries leading to functional equations for Stokes multipliers equivalent to
discrete integrable Hirota-type equations. The separation of "time" and "space"
variables, a consequence of our integrable structure, also leads to
Schr\"odinger ODEs and thus may have a connection with ODE/IM methods.
