The biological pumping of carbon in the ocean and its effects on ocean carbon sequestration are being studied by researchers from several disciplines. These studies address the biological carbon pump (also called organic, soft-tissue, or biogeochemical carbon pump), the carbonate pump (or counter-pump) and the microbial carbon pump, which are most often treated separately. In the present study, the three pumps are treated together for the first time, under the generic name of "biologically-driven ocean carbon pumps", in short, "bio-pumps". The interactions among the carbon fluxes of three interwoven bio-pumps are summarized in the bio-pump jigsaw puzzle. The bio-pump literature presents a wide range of approaches to the nature and processes of carbon pumps and the ensuing carbon sequestration in the ocean, approaches that often differ significantly. It is shown that sequestration fluxes from all three bio-pumps can occur throughout the water column, albeit in different forms, and this "continuous vertical sequestration" concept is used to propose a target framework to simplify the research on bio-pumps, and unify the studies carried out by researchers from different disciplines. The review of the wide range of approaches to ocean biologically-driven carbon pumping and sequestration in the literature includes both a look back at the initial ocean carbon pump concept and an analysis of current approaches. It also includes estimates of century-scale (≥100 years) global sequestration fluxes in the water column by the three bio-pumps, which are about 1–3, 0.7 and 0.2 Pg C y^–1 for the biological, carbonate and microbial pumps, respectively. The value of 0.7 Pg C y^–1 appears to be the first ever published for the carbonate pump. The review is followed by detailed analysis of the functioning of the bio-pumps and their carbon sequestration processes, which is organized around four common components, i.e. downward fluxes of biogenic carbon from the upper ocean (i.e. export), transformation fluxes of the exported biogenic carbon in the lower ocean, carbon sequestration fluxes throughout the water column, and upward flux of dissolved inorganic carbon. It is recommended that future carbon-pump publications use representations that include these four components. This leads to a synthetic description of the processes involved in the three bio-pumps and their carbon sequestration, and a proposed unification of research on the interwoven bio-pumps.