Model-theoretic results on ordinary and partial differential fields

n ordinary differential field is a field equipped with a single derivative. In 1959, A. Robinson defined the theory DCF of "differentially closed" ordinary differential fields of characteristic 0, and showed that DCF is complete and model complete. In 1968, L. Blum analyzed DCF from the viewpoint of stability theory, and in the process also found nice axioms.;In the first part of this dissertation we consider differential equations over ordinary differential fields (perhaps differentially closed) for which the set of solutions is strongly minimal. Hrushovski and Sokolovic show that the class of strongly minimal subsets of a differentially closed field are classified according to a certain trichotomy: "non-locally modular", "locally modular and non-trivial", or "locally modular and trivial." We show that for a certain class of such equations, the set of solutions is trivial. We also prove a theorem which provides a test for the orthogonality of types over an ordinary differential field.;In the second part of this dissertation we consider partial differential fields. A partial differential field is a field equipped with finitely many commuting derivations. We give axioms for m-DCF, the theory of differentially closed partial differential fields with m derivations, and show that this theory is complete and model complete. We show connections between model-theoretic properties of m-DCF and results in differential algebra. As a consequence we are able to perform stability-theoretic rank computations, and show that the theory m-DCF has rank...