The atmospheres of the four giant planets of our Solar System share a common and well-observed characteristic: they each display patterns of planetary banding, with regions of different temperatures, composition, aerosol properties and dynamics separated by strong meridional and vertical gradients in the zonal (i.e., east-west) winds. Remote sensing observations, from both visiting spacecraft and Earth-based astronomical facilities, have revealed the significant variation in environmental conditions from one band to the next. On Jupiter, the reflective white bands of low temperatures, elevated aerosol opacities, and enhancements of quasi-conserved chemical tracers are referred to as `zones.' Conversely, the darker bands of warmer temperatures, depleted aerosols, and reductions of chemical tracers are known as `belts.' On Saturn, we define cyclonic belts and anticyclonic zones via their temperature and wind characteristics, although their relation to Saturn's albedo is not as clear as on Jupiter. On distant Uranus and Neptune, the exact relationships between the banded albedo contrasts and the environmental properties is a topic of active study. This review is an attempt to reconcile the observed properties of belts and zones with (i) the meridional overturning inferred from the convergence of eddy angular momentum into the eastward zonal jets at the cloud level on Jupiter and Saturn and the prevalence of moist convective activity in belts; and (ii) the opposing meridional motions inferred from the upper tropospheric temperature structure, which implies decay and dissipation of the zonal jets with altitude above the clouds. These two scenarios suggest meridional circulations in opposing directions, the former suggesting upwelling in belts, the latter suggesting upwelling in zones. Numerical simulations successfully reproduce the former, whereas there is a wealth of observational evidence in support of the latter. This presents an unresolved paradox for our current understanding of the banded structure of giant planet atmospheres, that could be addressed via a multi-tiered vertical structure of "stacked circulation cells," with a natural transition from zonal jet pumping to dissipation as we move from the convectively-unstable mid-troposphere into the stably-stratified upper troposphere.