Considerable progress towards the achievement of thermonuclear burn using inertial confinement fusion has been achieved at the National Ignition Facility in the USA in the last few years. Other drivers, such as the Z-machine at Sandia, are also making progress towards this goal. A burning thermonuclear plasma would provide a unique and extreme plasma environment; in this paper we discuss (a) different theoretical challenges involved in modelling burning plasmas not currently considered, (b) the use of novel machine learning-based methods that might help large facilities reach ignition, and (c) the connections that a burning plasma might have to fundamental physics, including quantum electrodynamics studies, and the replication and exploration of conditions that last occurred in the first few minutes after the Big Bang. This article is part of a discussion meeting issue ‘Prospects for high gain inertial fusion energy (part 1)’.