  Adjoint functor theorems give necessary and sufficient conditions for a
functor to admit an adjoint. In this paper we prove general adjoint functor
theorems for functors between $\infty$-categories. One of our main results is
an $\infty$-categorical generalization of Freyd's classical General Adjoint
Functor Theorem. As an application of this result, we recover Lurie's adjoint
functor theorems for presentable $\infty$-categories. We also discuss the
comparison between adjunctions of $\infty$-categories and homotopy adjunctions,
and give a treatment of Brown representability for $\infty$-categories based on
Heller's purely categorical formulation of the classical Brown representability
theorem.
