We study domain-representable spaces, i.e., spaces that can be represented as the space of maximal elements of some continuous directed-complete partial order (= domain) with the Scott topology. We show that the Michael and Sorgenfrey lines are of this type, as is any subspace of any space of ordinals. We show that any completely regular space is a closed subset of some domain-representable space, and that if $X$ is domain-representable, then so is any $G_\delta $-subspace of $X$. It follows that any Čech-complete space is domain-representable. These results answer several questions in the literature.