After years of informal efforts, the parties to the United Nations Convention on the Law of the Sea (UNCLOS) are negotiating an international legally binding instrument to address governance gaps that have impeded attempts to conserve biodiversity in areas beyond national jurisdiction (ABNJ). Though these discussions were initiated in response to concerns about biodiversity, states have used them to advance competing claims regarding rights of access, ownership, and control of ocean spaces and resources. This paper examines how states have discussed ocean space in negotiations regarding area-based management tools (ABMTs) and marine genetic resources (MGRs) at the first three intergovernmental conferences regarding biodiversity in ABNJ. ABMTs, which have become widespread in governing ocean space for conservation, are premised on an ontological framing that ocean space can be divided by geographical boundaries into territories for management. MGRs, on the other hand, are newly recognized governance objects that cross existing spatial boundaries: between areas of national and international jurisdiction, between the seafloor and water column, and between the ocean and the laboratory. Through their mobility, MGRs reveal how territorial forms of governance over material resources intersect with other forms of exclusion, control, and rights-based institutions, suggesting the need to develop creative management regimes that go beyond territorial approaches.