A general-purpose robot should be able to master a wide range of tasks and quickly learn a novel one by leveraging past experiences. One-shot imitation learning (OSIL) approaches this goal by training an agent with (pairs of) expert demonstrations, such that at test time, it can directly execute a new task from just one demonstration. However, so far this framework has been limited to training on many variations of one task, and testing on other unseen but similar variations of the same task. In this work, we push for a higher level of generalization ability by investigating a more ambitious multi-task setup. We introduce a diverse suite of vision-based robot manipulation tasks, consisting of 7 tasks, a total of 61 variations, and a continuum of instances within each variation. For consistency and comparison purposes, we first train and evaluate single-task agents (as done in prior few-shot imitation work). We then study the multi-task setting, where multi-task training is followed by (i) one-shot imitation on variations within the training tasks, (ii) one-shot imitation on new tasks, and (iii) fine-tuning on new tasks. Prior state-of-the-art, while performing well within some single tasks, struggles in these harder multi-task settings. To address these limitations, we propose MOSAIC (Multi-task One-Shot Imitation with self-Attention and Contrastive learning), which integrates a self-attention model architecture and a temporal contrastive module to enable better task disambiguation and more robust representation learning. Our experiments show that MOSAIC outperforms prior state of the art in learning efficiency, final performance, and learns a multi-task policy with promising generalization ability via fine-tuning on novel tasks.