The voice conversion challenge is a bi-annual scientific event held to compare and understand different voice conversion (VC) systems built on a common dataset. In 2020, we organized the third edition of the challenge and constructed and distributed a new database for two tasks, intra-lingual semi-parallel and cross-lingual VC. After a two-month challenge period, we received 33 submissions, including 3 baselines built on the database. From the results of crowd-sourced listening tests, we observed that VC methods have progressed rapidly thanks to advanced deep learning methods. In particular, speaker similarity scores of several systems turned out to be as high as target speakers in the intra-lingual semi-parallel VC task. However, we confirmed that none of them have achieved human-level naturalness yet for the same task. The cross-lingual conversion task is, as expected, a more difficult task, and the overall naturalness and similarity scores were lower than those for the intra-lingual conversion task. However, we observed encouraging results, and the MOS scores of the best systems were higher than 4.0. We also show a few additional analysis results to aid in understanding cross-lingual VC better.