Virtually every connection to an Internet service is preceded by a DNS lookup. These lookups are performed in the clear without integrity protection, enabling manipulation, redirection, surveillance, and censorship. In parallel with standardization efforts that address these issues, large providers such as Google and Cloudflare are deploying solutions to encrypt lookups, such as DNS-over-TLS (DoT) and DNS-over-HTTPS (DoH). In this paper we examine whether encrypting DoH traffic can protect users from traffic analysis-based monitoring and censoring. We find that performing traffic analysis on DoH traces requires different features than those used to attack HTTPS or Tor traffic. We propose a new feature set tailored to the characteristics of DoH traffic. Our classifiers obtain an F1-score of 0.9 and 0.7 in closed and open world settings, respectively. We show that although factors such as location, resolver, platform, or client affect performance, they are far from completely deterring the attacks. We then study deployed countermeasures and show that, in contrast with web traffic, Tor effectively protects users. Specified defenses, however, still preserve patterns and leave some webs unprotected. Finally, we show that web censorship is still possible by analysing DoH traffic and discuss how to selectively block content with low collateral damage.