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Topological materials have in recent years become a subject of intense research due to fundamental considerations as well as potential use for technical applications in device fabrication and quantum information. One of the hallmarks of topological materials is the existence of protected exotic zero-energy surface states, which arise as a consequence of a nontrivial topology of the bulk wave functions. In this talk, starting from the ten-fold classification of topological insulators and superconductors, I will survey recent developments, with a particular emphasis on the topological classifications of fully gapped and gapless materials in terms of crystal reflection symmetries. As concrete examples, I discuss the Dirac materials Ca₃PbO and Sr₃PbO and show that these antiperovskites are reflection-symmetry-protected topological insulators. The Dirac surface states of these materials are protected by a non-zero mirror Chern number, which can take on only even values. As an example of a reflection-symmetry-protected topological semimetal, I will present results about the recently discovered compound Ca₃P₂ and demonstrate that this system exhibits surface states with a nearly flat dispersion. The latter implies that correlation effects are enhanced, which may lead to exotic symmetry-broken states at the surface of Ca₃P₂.