We report the first studies of luminescence spectra from asymmetric double quantum wells of GaAs with widths around 100 Å differing by only one (2.8 A) or two monolayers. Studies of the position and intensity of the direct and indirect exciton lines suggest the existence of acoustic phonon-assisted tunnelling between exciton states separated by a few meV. At temperatures 10-20 K the electron tunnelling process is fast enough to maintain thermal equilibrium between these direct and indirect excitons which are connected with holes in the same quantum well.PACS numbers: 73.20. Dx, 63.20.Kr In GaAs/AlGaAs double quantum wells (DQWs) electron and hole tunnelling through the thin barrier have been intensively studied by both cw and time-resolved luminescence techniques and it is clear that the tunnelling processes are signiflcantly modified by excitonic effects [1]. Much of this work was carried out on asymmetric DQWs in which the ground state energy differences in the two wells were appreciably more than 1 meV for both electrons and holes. For smaller energy differences, it has not yet been possible to separate the non-resonant tunnelling associated with acoustic phonons from that caused by elastic scattering from impurities and interface defects followed by phonon relaxation. For simplicity therefore we refer to the observed process as phonon-assisted tunnelling in the expectation that in our high quality samples this will turn out to be the stronger of the two. A theoretical analysis of these processes has been carried out [2,3] and a direct observation of acoustic phonon-assisted tunnelling in double barrier resonant tunnelling stuctures has been recently obtained using phonon pulses [4]. Detailed information on all these relaxation and tunnelling processes would be of value in the development of semiconductor nanostuctures as phonon spectrometers and generation at frequencies up to 1 ΤHz and above [2][3][4][5].In the present paper we are studying the cw photoluminescence from slightly asymmetric DQWs, in which the widths of the coupled QWs differ by only 1 or 2 monolayers (MLs). The paper is aimed particularly at extracting information on (895)