Recent advances in large language models (LLMs) have led to significant improvements in translating natural language questions into SQL queries. While achieving high accuracy in SQL generation is crucial, little is known about the extent to which these text-to-SQL models can reliably handle diverse types of questions encountered during real-world deployment, including unanswerable ones. To explore this aspect, we present TrustSQL, a new benchmark designed to assess the reliability of text-to-SQL models in both single-database and cross-database settings. The benchmark tasks models with providing one of two outcomes: 1) SQL prediction; or 2) abstention from making a prediction, either when there is a potential error in the generated SQL or when faced with unanswerable questions. For model evaluation, we explore various modeling approaches specifically designed for this task. These include: 1) optimizing separate models for answerability detection, SQL generation, and error detection, which are then integrated into a single pipeline; and 2) developing a unified approach that optimizes a single model to address the proposed task. Experimental results using our new reliability score show that addressing this challenge involves many different areas of research and opens new avenues for model development. Nonetheless, none of the methods surpass the reliability performance of the naive baseline, which abstains from answering all questions.