We present a discussion of Hubert et al. (2015)—hereafter, HRS15—that splits into two parts. In the first one, we argue that some structural properties of depth may, in some cases, limit its relevance for outlier detection. We also propose an alternative to bagdistances, which, while still based on depth, does not suffer from the same limitations. In the second part of the discussion, we investigate the possible uses of the weight functions that may enter the various integral functionals considered in HRS15.