We introduce deep neural networks for scanpath and saliency prediction trained on 360-degree images. The scanpath prediction model called SaltiNet is based on a temporal-aware novel representation of saliency information named the saliency volume. The first part of the network consists of a model trained to generate saliency volumes, whose parameters are fit by back-propagation using a binary cross entropy (BCE) loss over downsampled versions of the saliency volumes. Sampling strategies over these volumes are used to generate scanpaths over the 360-degree images. Our experiments show the advantages of using saliency volumes, and how they can be used for related tasks. We also show how a similar architecture achieves state-of-the-art performance for the related task of saliency map prediction. Our source code and trained models available here.