ASVspoof 2021 is the forth edition in the series of bi-annual challenges which aim to promote the study of spoofing and the design of countermeasures to protect automatic speaker verification systems from manipulation. In addition to a continued focus upon logical and physical access tasks in which there are a number of advances compared to previous editions, ASVspoof 2021 introduces a new task involving deepfake speech detection. This paper describes all three tasks, the new databases for each of them, the evaluation metrics, four challenge baselines, the evaluation platform and a summary of challenge results. Despite the introduction of channel and compression variability which compound the difficulty, results for the logical access and deepfake tasks are close to those from previous ASVspoof editions. Results for the physical access task show the difficulty in detecting attacks in real, variable physical spaces. With ASVspoof 2021 being the first edition for which participants were not provided with any matched training or development data and with this reflecting real conditions in which the nature of spoofed and deepfake speech can never be predicated with confidence, the results are extremely encouraging and demonstrate the substantial progress made in the field in recent years.