In the 1960s, Demazure and Grothendieck introduced the theory of reductive group schemes, a common generalization of the theories of compact Lie groups and Chevalley’s finite groups of Lie type. For many purposes in the Langlands program, it is useful to have a strong understanding of the behavior of reductive group schemes over general base schemes. Despite the extensive and fundamental work of Demazure—Grothendieck, there are still many fundamental questions left open. For instance, there are many constructions that are not known to be well-behaved; in technical terms, they are not known to be flat. My work focuses particularly on the theory of centralizers in reductive group schemes, a very common class of subgroup. Specifically, my work involves establishing the flatness of many different types of centralizers.