Determining which image regions to concentrate on is critical for Human-Object Interaction (HOI) detection. Conventional HOI detectors focus on either detected human and object pairs or pre-defined interaction locations, which limits learning of the effective features. In this paper, we reformulate HOI detection as an adaptive set prediction problem, with this novel formulation, we propose an Adaptive Set-based one-stage framework (AS-Net) with parallel instances and interaction branches. To attain this, we map a trainable interaction query set to an interaction prediction set with a transformer. Each query adaptively aggregates the interaction-relevant features from global contexts through multi-head co-attention. Besides, the training process is supervised adaptively by matching each ground truth with the interaction prediction. Furthermore, we design an effective instance-aware attention module to introduce instructive features from the instance branch into the interaction branch. Our method outperforms previous state-of-the-art methods without any extra human pose and language features on three challenging HOI detection datasets. Especially, we achieve over $31\%$ relative improvement on a large-scale HICO-DET dataset. Code is available at https://github.com/yoyomimi/AS-Net.