Abstract In this paper we study compactness of subsets of grand Lebesgue spaces (also known as Iwaniec–Sbordone spaces) and grand Sobolev spaces. Namely, we prove a Kolmogorov–Riesz compactness theorem for grand Lebesgue spaces and the corresponding version of a result of Sudakov. In addition, the validity of a Rellich–Kondrachov type compact embedding is shown to hold in the context of grand Sobolev spaces.