The archicortical hippocampus differs, like the neocortex, in its folding patterns between individuals. Here, we present an automated and robust BIDS-App, HippUnfold, for defining and indexing subject-specific hippocampal folding in MRI, analogous to popular tools used in neocortical reconstruction. This is critical for inter-individual alignment, with topology as the basis for homology. This topological framework enables qualitatively new analyses of morphological and laminar structure in the hippocampus or hippocampal subfields, and is critical for the advancement of neuroimaging analyses at a meso- or micro-scale. HippUnfold uses state-of-the-art deep learning combined with previously developed topological constraints on hippocampal tissue. It is designed to work with commonly employed sub-millimetric MRI acquisitions, with extensibility to microscopic resolutions as well. In this paper we illustrate the power of HippUnfold in feature extraction, and its construct validity compared to several extant hippocampal subfield analysis methods.