Aside from being spectacular displays in their own right, Type Ia  supernova explosions have a key role in measuring the expansion history  of the Universe and synthesizing the iron group elements.

But what is  their origin? That thermonuclear supernovae arise from exploding white dwarfs  is relatively well-established but the manner in which the explosion is ignited and  how this can be determined from what we observe remain hotly debated issues.

I will discuss the theoretical modelling of thermonuclear supernovae with particular focus on how our work on multi-dimensional radiative transfer simulations can be used to test  explosion scenarios. I will argue that understanding the diversity of thermonuclear supernovae  requires us to investigate a variety of different progenitor scenarios and review selected results  from our work on both Chandrasekhar-mass white-dwarf explosion models and sub-Chandrasekhar  scenarios.