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Recently network embedding has gained increasing attention due to its advantages in facilitating network computation tasks such as link prediction, node classification and node clustering. The objective of network embedding is to represent network nodes in a low-dimensional vector space while retaining as much information as possible from the original network including structural, relational, and semantic information. However, asymmetric nature of directed networks poses many challenges as how to best preserve edge directions in the embedding process. Here, we propose a novel deep asymmetric attributed network embedding model based on convolutional graph neural network, called AAGCN. The main idea is to maximally preserve the asymmetric proximity and asymmetric similarity of directed attributed networks. AAGCN introduces two neighbourhood feature aggregation schemes to separately aggregate the features of a node with the features of its in- and out- neighbours. Then, it learns two embedding vectors for each node, one source embedding vector and one target embedding vector. The final representations are the results of concatenating source and target embedding vectors. We test the performance of AAGCN on three real-world networks for network reconstruction, link prediction, node classification and visualization tasks. The experimental results show the superiority of AAGCN against state-of-the-art embedding methods.