The standard model of choice in economics is the maximization of a complete and transitive preference relation over a fixed set of alternatives. While completeness of preferences is usually regarded as a strong assumption, weakening it requires care to ensure that the resulting model still has enough structure to yield interesting results. This paper takes a step in this direction by studying the class of "connected preferences", that is, preferences that may fail to be complete but have connected maximal domains of comparability. We offer four new results. Theorem 1 identifies a basic necessary condition for a continuous preference to be connected in the sense above, while Theorem 2 provides sufficient conditions. Building on the latter, Theorem 3 characterizes the maximal domains of comparability. Finally, Theorem 4 presents conditions that ensure that maximal domains are arc-connected.