  This dissertation is a contribution to the project of second-order set
theory, which has seen a revival in recent years. The approach is to understand
second-order set theory by studying the structure of models of second-order set
theories. The main results are the following, organized by chapter. First, I
investigate the poset of $T$-realizations of a fixed countable model of
$\mathsf{ZFC}$, where $T$ is a reasonable second-order set theory such as
$\mathsf{GBC}$ or $\mathsf{KM}$, showing that it has a rich structure. In
particular, every countable partial order embeds into this structure. Moreover,
we can arrange so that these embedding preserve the existence/nonexistence of
upper bounds, at least for finite partial orders. Second I generalize some
constructions of Marek and Mostowski from $\mathsf{KM}$ to weaker theories.
They showed that every model of $\mathsf{KM}$ plus the Class Collection schema
"unrolls" to a model of $\mathsf{ZFC}^-$ with a largest cardinal. I calculate
the theories of the unrolling for a variety of second-order set theories, going
as weak as $\mathsf{GBC} + \mathsf{ETR}$. I also show that being $T$-realizable
goes down to submodels for a broad selection of second-order set theories $T$.
Third, I show that there is a hierarchy of transfinite recursion principles
ranging in strength from $\mathsf{GBC}$ to $\mathsf{KM}$. This hierarchy is
ordered first by the complexity of the properties allowed in the recursions and
second by the allowed heights of the recursions. Fourth, I investigate the
question of which second-order set theories have least models. I show that
strong theories---such as $\mathsf{KM}$ or $\Pi^1_1\text{-}\mathsf{CA}$---do
not have least transitive models while weaker theories---from $\mathsf{GBC}$ to
$\mathsf{GBC} + \mathsf{ETR}_\mathrm{Ord}$---do have least transitive models.
