Natural Language Generation (NLG) has improved exponentially in recent years thanks to the development of sequence-to-sequence deep learning technologies such as Transformer-based language models. This advancement has led to more fluent and coherent natural language generation, leading to improved development in downstream tasks such as abstractive summarization, dialogue generation and data-to-text generation. However, it is also apparent that deep learning based generation is prone to hallucinate unintended texts, which degrades the system performance and fail to meet user expectations in many real-world scenarios. In order to address this issue, there have been studies in measuring and mitigating hallucinated texts. However there has not been a comprehensive review of the state-of-the-art in hallucination detection and mitigation.In this survey, we provide a broad overview of the research progress and challenges in the hallucination problem of NLG. The survey is organized into two parts: (1) a general overview of metrics, mitigation methods, and future directions; (2) an overview of task-specific research progress for hallucinations in a large set of downstream tasks, namely abstractive summarization, dialogue generation, generative question answering, data-to-text generation, and machine translation. This survey serves to facilitate collaborative efforts among researchers in tackling the challenge of hallucinated texts in NLG.