  The targets of this article are threefold. The first one is to give a survey
on the recent developments of function spaces with mixed norms, including mixed
Lebesgue spaces, iterated weak Lebesgue spaces, weak mixed-norm Lebesgue spaces
and mixed Morrey spaces as well as anisotropic mixed-norm Hardy spaces. The
second one is to provide a detailed proof for a useful inequality about mixed
Lebesgue norms and the Hardy--Littlewood maximal operator and also to improve
some known results on the maximal function characterizations of anisotropic
mixed-norm Hardy spaces and the boundedness of Calder\'on--Zygmund operators
from these anisotropic mixed-norm Hardy spaces to themselves or to mixed
Lebesgue spaces. The last one is to correct some errors and seal some gaps
existing in the known articles.
