One of the most promising explanations for the origin of the billion solar mass black holes (BHs) inferred to power quasars at redshifts z >= 6 is that supermassive stars (SMSs) with masses >=104Msolar collapse to form the seed BHs from which they grow. Here we review recent theoretical advances which provide support for this scenario. Firstly, given sufficiently high accretion rates of gas into the cores of primordial protogalaxies, it appears that neither the high energy radiation emitted from the stellar surface nor the limited lifetime of SMSs can prevent their growth to masses of up to >=105Msolar. Secondly, recent cosmological simulations suggest that the high fluxes of molecule-dissociating radiation which may be required in order to achieve such high accretion rates may be more common in the early universe than previously thought. We conclude that the majority of supermassive BHs may originate from SMSs at high redshifts.