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In this paper, we tackle the task of establishing dense visual correspondences between images containing objects of the same category. This is a challenging task due to large intra-class variations and a lack of dense pixel level annotations. We propose a convolutional neural network architecture, called adaptive neighbourhood consensus network (ANC-Net), that can be trained end-to-end with sparse key-point annotations, to handle this challenge. At the core of ANC-Net is our proposed non-isotropic 4D convolution kernel, which forms the building block for the adaptive neighbourhood consensus module for robust matching. We also introduce a simple and efficient multi-scale self-similarity module in ANC-Net to make the learned feature robust to intra-class variations. Furthermore, we propose a novel orthogonal loss that can enforce the one-to-one matching constraint. We thoroughly evaluate the effectiveness of our method on various benchmarks, where it substantially outperforms state-of-the-art methods.