A permutation of size n can be identified with its diagram in which there is exactly one point in each row and column in the grid [n]². In this paper we consider multidimensional permutations (or d-permutations), which are identified with their diagrams on the grid [n]^d in which there is exactly one point per hyperplane x_i = j for i ∈ [d] and j ∈ [n]. We first exhaustively investigate all small pattern-avoiding classes for d = 3. We provide several bijections to enumerate some of these classes and we propose conjectures for others. We then give a generalization of the well-studied Baxter permutations to higher dimensions. In addition, we provide a vincular pattern-avoidance characterization of Baxter d-permutations.