Zero pronouns (ZPs) are frequently omitted in pro-drop languages (e.g. Chinese, Hungarian, and Hindi), but should be recalled in non-pro-drop languages (e.g. English). This phenomenon has been studied extensively in machine translation (MT), as it poses a significant challenge for MT systems due to the difficulty in determining the correct antecedent for the pronoun. This survey paper highlights the major works that have been undertaken in zero pronoun translation (ZPT) after the neural revolution so that researchers can recognize the current state and future directions of this field. We provide an organization of the literature based on evolution, dataset, method, and evaluation. In addition, we compare and analyze competing models and evaluation metrics on different benchmarks. We uncover a number of insightful findings such as: 1) ZPT is in line with the development trend of large language model; 2) data limitation causes learning bias in languages and domains; 3) performance improvements are often reported on single benchmarks, but advanced methods are still far from real-world use; 4) general-purpose metrics are not reliable on nuances and complexities of ZPT, emphasizing the necessity of targeted metrics; 5) apart from commonly-cited errors, ZPs will cause risks of gender bias.