The hippocampal-entorhinal system is important for spatial and relational memory tasks. We formally link these domains, provide a mechanistic understanding of the hippocampal role in generalization, and offer unifying principles underlying many entorhinal and hippocampal cell types. We propose medial entorhinal cells form a basis describing structural knowledge, and hippocampal cells link this basis with sensory representations. Adopting these principles, we introduce the Tolman-Eichenbaum machine (TEM). After learning, TEM entorhinal cells display diverse properties resembling apparently bespoke spatial responses, such as grid, band, border, and object-vector cells. TEM hippocampal cells include place and landmark cells that remap between environments. Crucially, TEM also aligns with empirically recorded representations in complex non-spatial tasks. TEM also generates predictions that hippocampal remapping is not random as previously believed; rather, structural knowledge is preserved across environments. We confirm this structural transfer over remapping in simultaneously recorded place and grid cells. Display omitted •Common principles for space and relational memory in the hippocampal formation•Explains hippocampal generalization in both spatial and non-spatial problems•Accounts for many reported hippocampal and entorhinal cell types from such tasks•Predicts how hippocampus remaps in both spatial and non-spatial tasks The Tolman-Eichenbaum Machine, named in honor of Edward Chace Tolman and Howard Eichenbaum for their contributions to cognitive theory, provides a unifying framework for the hippocampal role in spatial and nonspatial generalization and unifying principles underlying many entorhinal and hippocampal cell types.