Structured data sources, such as tables, graphs, and databases, are ubiquitous knowledge sources. Despite the demonstrated capabilities of large language models (LLMs) on plain text, their proficiency in interpreting and utilizing structured data remains limited. Our investigation reveals a notable deficiency in LLMs' ability to process structured data, e.g., ChatGPT lags behind state-of-the-art (SoTA) model by an average of 35%. To augment the Structured Knowledge Grounding (SKG) capabilities in LLMs, we have developed a comprehensive instruction tuning dataset comprising 1.1 million examples. Utilizing this dataset, we train a series of models, referred to as StructLM, based on the Mistral and the CodeLlama model family, ranging from 7B to 34B parameters. Our StructLM series surpasses task-specific models on 16 out of 18 evaluated datasets and establishes new SoTA performance on 8 SKG tasks. Furthermore, StructLM demonstrates strong generalization across 6 novel held-out SKG tasks, outperforming TableLlama by an average of 35\% and Flan-UL2 20B by an average of 10\%. Contrary to expectations, we observe that scaling model size offers marginal benefits, with StructLM-34B showing only slight improvements over StructLM-7B. This suggests that structured knowledge grounding is still a challenging task and requires more innovative design to push to a new level.