  We construct compactifications for median spaces with compact intervals,
generalising Roller boundaries of ${\rm CAT}(0)$ cube complexes. Examples of
median spaces with compact intervals include all finite rank median spaces and
all proper median spaces of infinite rank. Our methods also work for general
median algebras, where we recover the zero-completions of Bandelt and Meletiou.
Along the way, we prove various properties of halfspaces in finite rank median
spaces and a duality result for locally convex median spaces.
