The problem of convection in a fluid-saturated porous medium is reviewed with a focus on ‘vigorous’ convective flow, when the driving buoyancy forces are large relative to any dissipative forces in the system. This limit of strong convection is applicable in numerous settings in geophysics and beyond, including geothermal circulation, thermohaline mixing in the subsurface and heat transport through the lithosphere. Its manifestations range from ‘black smoker’ chimneys at mid-ocean ridges to salt-desert patterns to astrological plumes, and it has received a great deal of recent attention because of its important role in the long-term stability of geologically sequestered CO 2 . In this review, the basic mathematical framework for convection in porous media governed by Darcy’s Law is outlined, and its validity and limitations discussed. The main focus of the review is split between ‘two-sided’ and ‘one-sided’ systems: the former mimics the classical Rayleigh–Bénard set-up of a cell heated from below and cooled from above, allowing for detailed examination of convective dynamics and fluxes; the latter involves convection from one boundary only, which evolves in time through a series of regimes. Both set-ups are reviewed, accounting for theoretical, numerical and experimental studies in each case, and studies that incorporate additional physical effects are discussed. Future research in this area and various associated modelling challenges are also discussed.