We investigate set-theoretic dividing lines in model theory. In particular, we are interested in Keisler's order and Borel complexity. Keisler's order is a pre-order on complete countable theories T , measuring the saturation of ultrapowers of models of T . In Chapter 3, we present a self-contained survey on Keisler's order. In Chapter 4, we uniformize and sharpen several ultrafilter constructions of Malliaris and Shelah. We also investigate the model-theoretic properties detected by Keisler's

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... rder among the simple unstable theories. Borel complexity is a pre-order on sentences of L ω 1 ω measuring the complexity of countable models. In Chapter 5, we describe joint work with Richard Rast and Chris Laskowski on this order. In particular, we connect the Borel complexity of Φ ∈ L ω 1 ω with the number of potential canonical Scott sentences of Φ. In Chapter 6, we introduce the notion of thickness; when Φ has class-many potential canonical Scott sentences, thickness is a measure of how quickly this class grows in size. In Chapter 7, we describe joint work with Saharon Shelah on the Borel complexity of torsion-free abelian groups.