The discovery of gravitational waves, first observed in September 2015 following the merger of a binary black hole system, has already revolutionised our understanding of the Universe. This was further enhanced in August 2017, when the coalescence of a binary neutron star system was observed both with gravitational waves and a variety of electromagnetic counterparts; this joint observation marked the beginning of gravitational multimessenger astronomy. The Einstein Telescope, a proposed next-generation ground-based gravitational-wave observatory, will dramatically increase the sensitivity to sources: the number of observations of gravitational waves is expected to increase from roughly 100 per year to roughly 100’000 per year, and signals may be visible for hours at a time, given the low frequency cutoff of the planned instrument. This increase in the number of observed events, and the duration with which they are observed, is hugely beneficial to the scientific goals of the community but poses a number of significant computing challenges. Moreover, the currently used computing algorithms do not scale to this new environment, both in terms of the amount of resources required and the speed with which each signal must be characterised. This contribution will discuss the Einstein Telescope's computing challenges, and the activities that are underway to prepare for them. Available computing resources and technologies will greatly evolve in the years ahead, and those working to develop the Einstein Telescope data analysis algorithms will need to take this into account. It will also be important to factor into the initial development of the experiment's computing model the availability of huge parallel HPC systems and ubiquitous Cloud computing; the design of the model will also, for the first time, include the environmental impact as one of the optimisation metrics.