Many standard seismic data processing and imaging techniques require regularly sampled data. Prestack seismic data are multidimensional signals that can be represented via low-rank fourth-order tensors in the domain. We propose to adopt tensor completion strategies to recover unrecorded observations and to improve the signal-to-noise ratio of prestack seismic volumes. Tensor completion can be posed as an inverse problem and solved by minimizing a convex objective function. The objective function contains two terms: a data misfit and a nuclear norm. The data misfit measures the proximity of the reconstructed seismic data to the observations. The nuclear norm constraints the reconstructed data to be a low-rank tensor. In essence, we solve the prestack seismic reconstruction problem via low-rank tensor completion. The cost function of the problem is minimized using the alternating direction method of multipliers. We present synthetic examples to illustrate the behavior of the algorithm in terms of trade-off parameters that control the quality of the reconstruction. We further illustrate the performance of the algorithm with a land data survey from Alberta, Canada.