In this talk I will discuss various aspects of rational and integral structures on spaces of automorphic forms of reductive groups originating in the cohomology of arithmetic groups. A key concept behind the construction of rational and integral structures is the notion of locally algebraic representation which gives rise to locally algebraic g,K-modules as well as to associated sheaves on associated arithmetic orbifolds. These notions overcome certain conceptual obstacles in the standard treatment of arithmetic groups, their cohomology, and associated automorphic representations and align with Langlands' philosophy of parametrizing L-functions by L-group representations.