The use of industrial waste and/or by-products as alternative sources of raw materials in building materials has become standard practice. The result, more sustainable construction, is contributing to the institution of a circular economy. Nonetheless, all necessary precautions must be taken to ensure that the inclusion and use of such materials entail no new health hazard for people or their environment. Due to the processes involved in generating industrial waste/by-products, these alternative or secondary materials may be contaminated with heavy metals, other undesirable chemicals or high levels of natural radioactivity that may constrain their use. In-depth and realistic research on such industrial waste is consequently requisite to its deployment in building materials. This paper reviews the basic concepts associated with radioactivity and natural radioactivity, focusing on industrial waste/by-products comprising Naturally Occurring Radioactive Materials (NORM) used in cement and concrete manufacture. Updated radiological data are furnished on such waste (including plant fly ash, iron and steel mill slag, bauxite and phosphogypsum waste) and on other materials such as limestone, gypsum and so on. The paper also presents recent findings on radionuclide activity concentrations in Portland cements and concretes not bearing NORMs. The role of natural aggregate in end concrete radiological behaviour is broached. The radiological behaviour of alternative non-portland cements and concretes, such as alkali-activated materials and geopolymers, is also addressed.