Over the coming century humanity may need to find reservoirs to store several trillions of tons of carbon dioxide (CO2) emitted from fossil fuel combustion, which would otherwise cause dangerous climate change if it were left in the atmosphere. Carbon storage in the ocean as bicarbonate ions (by increasing ocean alkalinity) has received very little attention. Yet recent work suggests sufficient capacity to sequester copious quantities of CO2. It may be possible to sequester hundreds of billions to trillions of tons of C without surpassing postindustrial average carbonate saturation states in the surface ocean. When globally distributed, the impact of elevated alkalinity is potentially small and may help ameliorate the effects of ocean acidification. However, the local impact around addition sites may be more acute but is specific to the mineral and technology. The alkalinity of the ocean increases naturally because of rock weathering in which >1.5 mol of carbon are removed from the atmosphere for every mole of magnesium or calcium dissolved from silicate minerals (e.g., wollastonite, olivine, and anorthite) and 0.5 mol for carbonate minerals (e.g., calcite and dolomite). These processes are responsible for naturally sequestering 0.5 billion tons of CO2 per year. Alkalinity is reduced in the ocean through carbonate mineral precipitation, which is almost exclusively formed from biological activity. Most of the previous work on the biological response to changes in carbonate chemistry have focused on acidifying conditions. More research is required to understand carbonate precipitation at elevated alkalinity to constrain the longevity of carbon storage. A range of technologies have been proposed to increase ocean alkalinity (accelerated weathering of limestone, enhanced weathering, electrochemical promoted weathering, and ocean liming), the cost of which may be comparable to alternative carbon sequestration proposals (e.g., $20–100 tCO2−1). There are still many unanswered technical, environmental, social, and ethical questions, but the scale of the carbon sequestration challenge warrants research to address these.