I will begin with a friendly introduction to the deformation theory of Galois representations and its role in modularity lifting, focusing on the case of elliptic curves over Q. This will motivate the study of local deformation rings and more specifically flat deformations. Next, we will discuss Kisin’s resolution of the flat deformation ring at l = p and describe conceptually the importance of local models of Shimura varieties in analyzing its geometry. In the remaining time, we will address the title of the talk; the additional structure we consider could be a symplectic form, an orthogonal form, or more generally any reductive subgroup G of $GL_N$. I will describe briefly the role that recent advances in p-adic Hodge theory and local models of Shimura varieties play in this situation.