This review provides an introduction to the field of topological quantum matter, focusing on classification schemes. It covers both fully gapped and gapless topological materials, emphasizing nonspatial symmetries like time-reversal and spatial symmetries like reflection. The classification of gapless modes localized on topological defects is also discussed using homotopy groups, Clifford algebras, K-theory, and non-linear sigma models. The review includes theoretical model systems and experimental results to provide a comprehensive perspective on the field. While the main focus is on noninteracting or mean-field Hamiltonians, the article also touches on recent results and open questions regarding interacting systems. The content is organized into sections on symmetries, fully gapped free fermion systems, and gapless materials, with a brief overview of interacting SPT phases. The review aims to serve as a reference for researchers studying topological quantum matter.This review provides an introduction to the field of topological quantum matter, focusing on classification schemes. It covers both fully gapped and gapless topological materials, emphasizing nonspatial symmetries like time-reversal and spatial symmetries like reflection. The classification of gapless modes localized on topological defects is also discussed using homotopy groups, Clifford algebras, K-theory, and non-linear sigma models. The review includes theoretical model systems and experimental results to provide a comprehensive perspective on the field. While the main focus is on noninteracting or mean-field Hamiltonians, the article also touches on recent results and open questions regarding interacting systems. The content is organized into sections on symmetries, fully gapped free fermion systems, and gapless materials, with a brief overview of interacting SPT phases. The review aims to serve as a reference for researchers studying topological quantum matter.