  The classification problem of $P$- and $Q$-polynomial association schemes has
been one of the central problems in algebraic combinatorics. Generalizing the
concept of $P$- and $Q$-polynomial association schemes to multivariate cases,
namely to consider higher rank $P$- and $Q$-polynomial association schemes, has
been tried by some authors, but it seems that so far there were neither very
well-established definition nor results. Very recently, Bernard, Cramp\'{e},
d'Andecy, Vinet, and Zaimi [arXiv:2212.10824], defined bivariate $P$-polynomial
association schemes, as well as bivariate $Q$-polynomial association schemes.
In this paper, we study these concepts and propose a new modified definition
concerning a general monomial order, which is more general and more natural and
also easy to handle. We prove that there are many interesting families of
examples of multivariate $P$- and/or $Q$-polynomial association schemes.
