  A permutation of size $n$ can be identified to its diagram in which there is
exactly one point per row and column in the grid $[n]^2$. In this paper we
consider multidimensional permutations (or $d$-permutations), which are
identified to their diagrams on the grid $[n]^d$ in which there is exactly one
point per hyperplane $x_i=j$ for $i\in[d]$ and $j\in[n]$. We first investigate
exhaustively all small pattern avoiding classes. We provide some bijection to
enumerate some of these classes and we propose some conjectures for others. We
then give a generalization of well-studied Baxter permutations into this
multidimensional setting. In addition, we provide a vincular pattern avoidance
characterization of Baxter $d$-permutations.
