The ASVspoof initiative was conceived to spearhead research in anti-spoofing for automatic speaker verification (ASV). This paper describes the third in a series of biannual challenges: ASVspoof 2019. With the challenge database and protocols being described elsewhere, the focus of this paper is on results and the top performing single and ensemble system submissions from 62 teams, all of which out-perform the two baseline systems, often by a substantial margin. Deeper analyses shows that performance is dominated by specific conditions involving either specific spoofing attacks or specific acoustic environments. While fusion is shown to be particularly effective for the logical access scenario involving speech synthesis and voice conversion attacks, participants largely struggled to apply fusion successfully for the physical access scenario involving simulated replay attacks. This is likely the result of a lack of system complementarity, while oracle fusion experiments show clear potential to improve performance. Furthermore, while results for simulated data are promising, experiments with real replay data show a substantial gap, most likely due to the presence of additive noise in the latter. This finding, among others, leads to a number of ideas for further research and directions for future editions of the ASVspoof challenge.