Title: Motif-induced Graph Normalization. Abstract: Graph Neural Networks (GNNs) have emerged as a powerful category of learning architecture for handling graph-structured data in the non-Euclidean domain. Despite their success, existing GNNs typically suffer from the insufficient expressive power bottlenecked by Weisfeiler-Lehman (WL) test, and meanwhile are prone to the over-smoothing situation with increasing layer numbers. In this paper, we strive to strengthen the discriminative capabilities of GNNs by devising a dedicated plug-and-play normalization scheme, termed as Motif-induced Normalization (MotifNorm), that explicitly considers the intra-connection information within each node-induced subgraph. To this end, we embed the motif-induced structural weights at the beginning and the end of the standard BatchNorm, as well as incorporate the graph instance-specific statistics for improved distinguishable capabilities. In the meantime, we provide the theoretical analysis to support that, with the proposed elaborated MotifNorm, an arbitrary GNNs is capable of more expressive abilities than the 1-WL test in distinguishing k-regular graphs. Furthermore, the proposed MotifNorm scheme is also exemplified to be able to alleviate the over-smoothing phenomenon. Experimental results on ten popular benchmarks across all the tasks of the graph-, node-, as well as link-level property predictions, demonstrate the effectiveness of the proposed method. Our code is made available in the supplementary material.