Unsupervised transfer of object recognition models from synthetic to real data is an important problem with many potential applications. The challenge is how to "adapt" a model trained on simulated images so that it performs well on real-world data without any additional supervision. Unfortunately, current benchmarks for this problem are limited in size and task diversity. In this paper, we present a new large-scale benchmark called Syn2Real, which consists of a synthetic domain rendered from 3D object models and two real-image domains containing the same object categories. We define three related tasks on this benchmark: closed-set object classification, open-set object classification, and object detection. Our evaluation of multiple state-of-the-art methods reveals a large gap in adaptation performance between the easier closed-set classification task and the more difficult open-set and detection tasks. We conclude that developing adaptation methods that work well across all three tasks presents a significant future challenge for syn2real domain transfer.