Multimode fibres (MMF) are remarkable high-capacity information channels owing to the large number of transmitting fibre modes, and have recently attracted significant renewed interest in applications such as optical communication, imaging, and optical trapping. At the same time, the optical transmitting modes inside MMFs are highly sensitive to external perturbations and environmental changes, resulting in MMF transmission channels being highly variable and random. This largely limits the practical application of MMFs and hinders the full exploitation of their information capacity. Despite great research efforts made to overcome the high variability and randomness inside MMFs, any geometric change to the MMF leads to completely different transmission matrices, which unavoidably fails at the information recovery. Here, we show the successful binary image transmission using deep learning through a single MMF, which is stationary or subject to dynamic shape variations. We found that a single convolutional neural network has excellent generalisation capability with various MMF transmission states. This deep neural network can be trained by multiple MMF transmission states to accurately predict unknown information at the other end of the MMF at any of these states, without knowing which state is present. Our results demonstrate that deep learning is a promising solution to address the variability and randomness challenge of MMF based information channels. This deep-learning approach is the starting point of developing future high-capacity MMF optical systems and devices, and is applicable to optical systems concerning other diffusing media.