We study the stability of disc accretion in the recurrent nova RS Ophiuchi. We construct a one-dimensional time-dependent model of the binary–disc system, which includes viscous heating and radiative cooling and a self-consistent treatment of the binary potential. We find that the extended accretion disc in this system is always unstable to the thermal–viscous instability, and undergoes repeated disc outbursts on ∼10–20 yr time-scales. This is similar to the recurrence time-scale of observed outbursts in the RS Oph system, but we show that the disc’s accretion luminosity during outburst is insufficient to explain the observed outbursts. We explore a range of models, and find that in most cases the accretion rate during outbursts reaches or exceeds the critical accretion rate for stable nuclear burning on the white dwarf surface. Consequently we suggest that a surface nuclear burning triggered by disc instability may be responsible for the observed outbursts. This allows the white dwarf mass to grow over time, and we suggest that disc instability in RS Oph and similar systems may represent a path to Type Ia supernovae.