In this talk we will step back from the generality of multidimensional shifts of finite type and focus on two specific classes: The space of graph homomorphisms from Zd to a fixed connected undirected graph and the space of tilings of Zd by rectangular shapes. While these classes cover fairly well-studied examples like the space of proper 3-colourings and domino tilings, a lot still remains to be understood. We will discuss some recent results about these shift spaces and try to pinpoint the difficulties which arise in their investigation.