Scaling phenomena, or self-similarity, are pervasive in nature and in data, and have been the subject of decades of research in probability and statistics. In higher dimension, self-similarity presents new challenges. These include the theoretical consequences of matrix-scaling, anisotropy, non-identifiability, and their impact on inferential pursuits. In this talk, we will give a broad view of related probabilistic and inferential issues in multidimensional settings. We will describe recent developments for multivariate, multiparameter Gaussian self-similar random fields, the so-named operator fractional Brownian fields (OF-BFs). The analysis will draw upon harmonizable integral representations; the latter will allow us to characterize the symmetry groups and anisotropy of OFBFs. We will also discuss recent efforts on wavelet-based inference for multivariate self-similarity.