The effects of capillary forces on the propagation of two-phase, constant-flux gravity currents in a porous medium are studied analytically and numerically in an axisymmetric geometry. The fluid within a two-phase current generally only partially saturates the pore space it invades. For long, thin currents, the saturation distribution is set by the vertical balance between gravitational and capillary forces. The capillary pressure and relative permeability of the fluid in the current depend on this saturation. The action of capillary forces reduces the average saturation, thereby decreasing the relative permeability throughout the current. This results in a thicker current, which provides a steeper gradient to drive flow, and a more blunt-nose profile. The relative strength of gravity and capillary forces remains constant within a two-phase gravity current fed by a constant flux and spreading radially, due to mass conservation. For this reason, we use an axisymmetric representation of the framework developed by Golding et al. ["Two-phase gravity currents in porous media," J. Fluid Mech.678, 248-270 (2011)]10.1017/jfm.2011.110, to investigate the effect on propagation of varying the magnitude of capillary forces and the pore-size distribution. Scaling analysis indicates that axisymmetric two-phase gravity currents fed by a constant flux propagate like t1/2, similar to their single-phase counterparts [S. Lyle, H. E. Huppert, M. Hallworth, M. Bickle, and A. Chadwick, "Axisymmetric gravity currents in a porous medium," J. Fluid Mech.543, 293-302 (2005)]10.1017/S0022112005006713, with the effects of capillary forces encapsulated in the constant of proportionality. As a practical application of our new concepts and quantitative evaluations, we discuss the implications of our results for the process of carbon dioxide (CO2) sequestration, during which gravity currents consisting of supercritical CO2 propagate in rock saturated with aqueous brine. We apply our two-phase model including capillary forces to quantitatively assess seismic images of CO2 spreading at Sleipner underneath the North Sea.