Shimura varieties are moduli spaces of abelian varieties with extra structures (e.g. algebraic cycles, or more generally Hodge cycles). Over the decades, various mathematicians (e.g. Mumford, Deligne, Rapoport, Kottwitz, etc.) have constructed nice integral models of Shimura varieties. In this talk, I will discuss some motivic aspects of integral models of Hodge type (or more generally abelian type) constructed by Kisin and Kisin-Pappas. I will talk about my recent work on removing the normalization step in the construction of such integral models, which gives closed embeddings of Hodge type integral models into Ag. I will also mention an application to toroidal compactifications of such integral models. 

If time permits, I will also mention a new result on connected components of affine Deligne–Lusztig varieties, which gives us a new CM (i.e. complex multiplication) lifting result for integral models of Shimura varieties at parahoric levels and serves as an ingredient for my main theorem at parahoric levels.