  We develop domain theory in constructive and predicative univalent
foundations (also known as homotopy type theory). That we work predicatively
means that we do not assume Voevodsky's propositional resizing axioms. Our work
is constructive in the sense that we do not rely on excluded middle or the
axiom of (countable) choice. Domain theory studies so-called directed complete
posets (dcpos) and Scott continuous maps between them and has applications in
programming language semantics, higher-type computability and topology. A
common approach to deal with size issues in a predicative foundation is to work
with information systems, abstract bases or formal topologies rather than
dcpos, and approximable relations rather than Scott continuous functions. In
our type-theoretic approach, we instead accept that dcpos may be large and work
with type universes to account for this. A priori one might expect that complex
constructions of dcpos result in a need for ever-increasing universes and are
predicatively impossible. We show that such constructions can be carried out in
a predicative setting. We illustrate the development with applications in the
semantics of programming languages: the soundness and computational adequacy of
the Scott model of PCF and Scott's $D_\infty$ model of the untyped
$\lambda$-calculus. We also give a predicative account of continuous and
algebraic dcpos, and of the related notions of a small basis and its rounded
ideal completion. The fact that nontrivial dcpos have large carriers is in fact
unavoidable and characteristic of our predicative setting, as we explain in a
complementary chapter on the constructive and predicative limitations of
univalent foundations. Our account of domain theory in univalent foundations is
fully formalised with only a few minor exceptions. The ability of the proof
assistant Agda to infer universe levels has been invaluable for our purposes.
