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After briefly recalling the quantum entanglement-based view of topologicalphases of matter in order to outline the general context, we give an overviewof different approaches to the classification problem of topological insulatorsand superconductors of non-interacting Fermions. In particular, we review insome detail general symmetry aspects of the "ten-fold way" which forms thefoundation of the classification, and put different approaches to theclassification in relationship with each other. We end by briefly mentioningsome of the results obtained on the effect of interactions, mainly in threespatial dimensions.

Topological phases: Classification of topological insulators and superconductors of non-interacting Fermions, and beyond