Turbulent buoyant jet diffusion flames have been investigated using large-eddy simulations of low Mach number compressible flows. Infinitely fast chemistry of Burke-Sehumann formulation was employed to explain the combustion process. The dynamic phenomena of flame flickering or puffing associated with the formation of regular large vortex structures in the near-field are captured and the puffing frequency is found to be weakly dependent on the shear-layer thickness. The Strouhal number at the non-dimensional puffing frequency agrees well with the experimental correlation. The processes of the transition from laminar to turbulent flow and the breakdown of the continuous flame into an intermittent flame zone followed by a plume-like zone are well reproduced. A quantitative description of the reacting flow field is given in terms of the mean, root-mean-square (r.m.s.) and probability density function (pdf) profiles of the axial velocity, temperature, species concentration, and mixture fraction. The decay of the centreline mean velocity obeys the -1/3 power law in terms of the streamwise distance in the plume-like region, which is consistent with the experimental observations. There exist inflection points in the mean and r.m.s. profiles in the near-field, which are caused by the existence of large-scale vortical structures and by chemical heat release. The pdfs of the temperature and mixture fraction attain a variety of shapes: Gaussian, single modal, bimodal, triangular, delta, which reflect the different flow and flame characteristics in different regions. Budgets of the mean axial momentum, radial momentum, and energy are analyzed in detail for the near field potential core region and the far field plume-like region. The buoyancy effect is large in the budget of the axial momentum and the pressure gradient plays a significant role in the radial momentum budget in both regions. Radial convection is strong in the near field, associated with the large-scale entrainment process. On the other hand, turbulent radial transport associated with turbulent mixing can have a large influence in both the near and the far fields.