  The Musielak--Orlicz setting unifies the variable exponent, Orlicz, weighted
Sobolev, and double-phase spaces. They inherit technical difficulties resulting
from general growth and inhomogeneity. In this survey we present an overview of
developments of the theory of existence of~PDEs in the setting including
reflexive and non-reflexive cases, as well as isotropic and anisotropic ones.
Particular attention is paid to problems with data below natural duality in
absence of Lavrentiev's phenomenon.
