The present work is concerned with the mathematical modelling of a bed of granular material in a gravitational field vertically fluidised by a vibrating surface. The particles are in rapid motion, and lose energy by inelastic collisions. The steady state is maintained by a balance of the rate of dissipation of energy in inelastic particle collisions and the rate of transfer of energy due to particle collisions with the vibrating surface. The limit where the energy dissipation due to inelastic collisions is small compared to the mean kinetic energy of the particles is considered. This non-equilibrium steady state is similar to a dilute gas at equilibrium with a uniform temperature and an exponentially decaying density, obtained from the ideal gas equation of state. From the analysis of this state, four non-dimensional numbers are derived which uniquely specify the state of the system. A perturbative analysis about the uniform temperature state is carried out and analytical solutions to the macroscopic variables of the system are obtained using two types of approximations. The first is a hydrodynamic model using constitutive relations from the general kinetic theory of granular media, and the second is a kinetic theory formulation derived exclusively for the vibro-fluidised bed. The latter permits an anisotropy between the horizontal and vertical directions due to the anisotropic nature of the source of energy at the bottom wall. The kinetic theory is extended to incorporate the corrections due to the high density effects, which is similar to the Enskog correction to dense gases. An event driven (ED), or hard sphere molecular dynamic (MD), simulation of the vibrated bed is carried out. The quantitative predictions of the theories are validated by the simulation. A systematic probing of the parameter space within the ED simulations revealed two new phenomena in a vibro-fluidised bed which are inhomogeneous in the horizontal direction. These are convection rolls similar to the Rayleigh-Benard instability in fluids, and a clustering instability leading to a phase separation. The instabilities are characterised using a phase diagram. The homogeneous states close to these new states are adequately described by the models developed here. An analysis of the stability of this state could have implications in understanding the instabilities in driven granular materials (such as in sheared media and fluidised beds) in general, and pattern formation in vibrated beds in particular.