The validity of a steady-flamelet model and a flamelet/progress-variable approach for gaseous and spray combustion is investigated by a two-dimensional direct numerical simulation (DNS) of gaseous and spray jet flames, and the combustion characteristics are analysed. A modified flamelet/progress-variable approach, in which total enthalpy rather than product mass fraction is chosen as a progress variable, is also examined. DNS with an Arrhenius formation, in which the chemical reaction is directly solved in the physical flow field, is performed as a reference to validate the combustion models. The results show that the diffusion flame is dominant in the gaseous diffusion jet flame, whereas diffusion and premixed flames coexist in the spray Jet flame. The characteristics of the spray flame change from premixed-diffusion coexistent to diffusion-dominant downstream. Comparisons among the results from DNS with various combustion models show the modified flamelet/progress-variable approach to be superior to the other combustion models, particularly for the spray flame. Where the behaviour of the gaseous total enthalpy is strongly affected by the energy transfer (i.e. heat transfer and mass transfer) from the dispersed droplet, and this effect can be accounted for only by solving the conservation equation of the total enthalpy. However, even the DNS with the modified flamelet/progress-variable approach tends to underestimate the gaseous temperature in the central region of the spray jet flame. To increase the prediction accuracy, a combustion model for the partially premixed flame for the spray flame is necessary.