Direct numerical simulation (DNS) of passive (non-buoyant) and active (buoyant) scalar homogeneous turbulence is carried out using a standard pseudo-spectral numerical method. The flow settings simulated include stationary forced and decaying passive-scalar turbulence, as well as decaying anisotropic active-scalar turbulence. The Schmidt number is unity in all cases. The results are compared with, and are found to be in very good agreement with, previous similar DNS studies. The well-validated DNS data are divided into 19 sets, and are employed to study different large eddy simulation (LES) subgrid-scale (SGS) models for the SGS scalar flux. The models examined include three eddy-viscosity-type models (Smagorinsky, Vreman and Sigma with a constant SGS Schmidt number), a Dynamic Structure model and two versions of the Gradient (Gradient and Modulated Gradient) model. The models are investigated with respect to their ability to predict the orientation, and the magnitude, of the SGS scalar flux. Eddy-viscosity models are found to predict the magnitude of the SGS scalar flux accurately, but are poor at predicting the orientation of the SGS scalar flux. The Dynamic Structure and Gradient models are better than eddy-viscosity models at predicting both the magnitude and direction. However, neither of them can be realised in an actual LES, without carrying additional transport equations. Based on these observations, four new models are proposed - combining directions from Dynamic Structure and Gradient models, and magnitudes from Smagorinsky and Vreman eddy-viscosity models. These models are expected to be better than eddy-viscosity and Modulated Gradient models, and this is confirmed by preliminary a posteriori tests. © 2014 Taylor & Francis.