This thesis concerns the study of affine Grassmannians and of local models for ramified triality groups. The triality groups we consider are groups of type 3D4, so they are forms of the orthogonal or the spin groups in 8 variables. They can be given as automorphisms of certain twisted composition algebras obtained from the octonion algebra. Using these composition algebras, we give descriptions of the affine Grassmannians and of the global affine Grassmannians for these triality groups as functors classifying suitable lattices in a fixed space. We combine these descriptions with the Pappas-Zhu construction, to obtain a corresponding description of local models for triality groups; the singularities of these models are supposed to model the singularities of certain orthogonal Shimura varieties.Moreover, we give a definition of a corresponding splitting model in terms of linear algebra data; this splitting model is expected to provide a partial resolution of the local model. By explicit calculations, we find equations that describe affine charts of the splitting model. Using these calculations, we show that the splitting model is isomorphic to the blow-up of a quadratic hypersurface along a specific smooth closed subscheme of its special fiber. It follows that the splitting model is regular and has special fiber which is the union of two smooth irreducible components that intersect transversely.
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