The previously unstudied problem of an axisymmetric gravity current released on a cone is investigated. A formulation of the problem based on the shallow-water equations, with and without entrainment and bottom drag, is given. Analytical asymptotic solutions are found and compared to numerical solutions of the equations. The inclusion of entrainment and drag is seen to play a significant role in limiting the propagation speed of the gravity current and also in altering the shape of the current. These theoretical predictions are compared to laboratory experiments and to two-dimensional numerical simulations. The shallow-water solutions including entrainment are found to provide a much better comparison with the experiments than the solutions without entrainment. In particular, the observed front speed and dilution of the head are in good agreement. Some applications to industrial and environmental problems such as dense gas dispersion are briefly discussed.