Question-answering tasks over structured knowledge (i.e., tables and graphs) require the ability to encode structural information. Traditional pre-trained language models trained on linear-chain natural language cannot be directly applied to encode tables and graphs. The existing methods adopt the pre-trained models in such tasks by flattening structured knowledge into sequences. However, the serialization operation will lead to the loss of the structural information of knowledge. To better employ pre-trained transformers for structured knowledge representation, we propose a novel structure-aware transformer (SATrans) that injects the local-to-global structural information of the knowledge into the mask of the different self-attention layers. Specifically, in the lower self-attention layers, SATrans focus on the local structural information of each knowledge token to learn a more robust representation of it. In the upper self-attention layers, SATrans further injects the global information of the structured knowledge to integrate the information among knowledge tokens. In this way, the SATrans can effectively learn the semantic representation and structural information from the knowledge sequence and the attention mask, respectively. We evaluate SATrans on the table fact verification task and the knowledge base question-answering task. Furthermore, we explore two methods to combine symbolic and linguistic reasoning for these tasks to solve the problem that the pre-trained models lack symbolic reasoning ability. The experiment results reveal that the methods consistently outperform strong baselines on the two benchmarks.