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A Natural Language Interface (NLI) enables the use of human languages to interact with computer systems, including smart phones and robots. Compared to other types of interfaces, such as command line interfaces (CLIs) or graphical user interfaces (GUIs), NLIs stand to enable more people to have access to functionality behind databases or APIs as they only require knowledge of natural languages. Many NLI applications involve structured data for the domain (e.g., applications such as hotel booking, product search, and factual question answering.) Thus, to fully process user questions, in addition to natural language comprehension, understanding of structured data is also crucial for the model. In this paper, we study neural network methods for building Natural Language Interfaces (NLIs) with a focus on learning structure data representations that can generalize to novel data sources and schemata not seen at training time. Specifically, we review two tasks related to natural language interfaces: i) semantic parsing where we focus on text-to-SQL for database access, and ii) task-oriented dialog systems for API access. We survey representative methods for text-to-SQL and task-oriented dialog tasks, focusing on representing and incorporating structured data. Lastly, we present two of our original studies on structured data representation methods for NLIs to enable access to i) databases, and ii) visualization APIs.