Passwords are undoubtedly the most dominant user authentication mechanism on the web today. Although they are inexpensive and easy-to-use, security concerns of password-based authentication are serious. Phishing and theft of password databases are two critical concerns. The tendency of users to re-use passwords across different services exacerbates the impact of these two concerns. Current solutions addressing these concerns are not fully satisfactory: they typically address only one of the two concerns; they do not protect passwords from rogue servers; they do not provide any verifiable evidence of their (server-side) adoption to users; and they face deployability challenges in terms of the cost for service providers and/or ease-of-use for end users. We present SafeKeeper, a comprehensive approach to protect the confidentiality of passwords in web authentication systems. Unlike previous approaches, SafeKeeper protects user passwords against very strong adversaries, including rogue servers and sophisticated external phishers. It is relatively inexpensive to deploy as it (i) uses widely available hardware security mechanisms like Intel SGX, (ii) is integrated into popular web platforms like WordPress, and (iii) has small performance overhead. We describe a variety of challenges in designing and implementing such a system, and how we overcome them. Through an 86-participant user study, and systematic analysis and experiments, we demonstrate the usability, security and deployability of SafeKeeper, which is available as open-source.