Graph neural networks (GNNs) and variations of the message passing algorithm are the predominant means for learning on graphs, largely due to their flexibility, speed, and satisfactory performance. The design of powerful and general purpose GNNs, however, requires significant research efforts and often relies on handcrafted, carefully-chosen message passing operators. Motivated by this, we propose a remarkably simple alternative for learning on graphs that relies exclusively on attention. Graphs are represented as node or edge sets and their connectivity is enforced by masking the attention weight matrix, effectively creating custom attention patterns for each graph. Despite its simplicity, masked attention for graphs (MAG) has state-of-the-art performance on long-range tasks and outperforms strong message passing baselines and much more involved attention-based methods on over 55 node and graph-level tasks. We also show significantly better transfer learning capabilities compared to GNNs and comparable or better time and memory scaling. MAG has sub-linear memory scaling in the number of nodes or edges, enabling learning on dense graphs and future-proofing the approach.