Hyperspectral (HS) image always suffers from the deficiency of low spatial resolution, compared with conventional optical image types, which has limited its further applications in remote sensing areas. Therefore, HS image super-resolution (SR) techniques are broadly employed in order to observe finer spatial structures while preserving the spectra of ground covers. In this paper, a novel multi-dimensional attention-aided transposed convolutional long-short term memory (LSTM) network is proposed for single HS image super-resolution task. The proposed network employs the convolutional bi-directional LSTM for the purpose of local and non-local spatial–spectral feature explorations, and transposed convolution for the purpose of image amplification and reconstruction. Moreover, a multi-dimensional attention module is proposed, aiming to capture the salient features on spectral, channel, and spatial dimensions, simultaneously, to further improve the learning abilities of network. Experiments on four commonly-used HS images demonstrate the effectiveness of this approach, compared with several state-of-the-art deep learning-based SR methods.