We propose a novel cost aggregation network, called Cost Aggregation with Transformers (CATs), to find dense correspondences between semantically similar images with additional challenges posed by large intra-class appearance and geometric variations. Compared to previous hand-crafted or CNN-based methods addressing the cost aggregation stage, which either lack robustness to severe deformations or inherit the limitation of CNNs that fail to discriminate incorrect matches due to limited receptive fields, CATs explore global consensus among initial correlation map with the help of some architectural designs that allow us to exploit full potential of self-attention mechanism. Specifically, we include appearance affinity modelling to disambiguate the initial correlation maps and multi-level aggregation to benefit from hierarchical feature representations within Transformer-based aggregator, and combine with swapping self-attention and residual connections not only to enforce consistent matching, but also to ease the learning process. We conduct experiments to demonstrate the effectiveness of the proposed model over the latest methods and provide extensive ablation studies. Code and trained models will be made available at https://github.com/SunghwanHong/CATs.