Virtually every connection to an Internet service is preceded by a DNS lookup. Lookups are performed without any traffic-level protection, thus enabling manipulation, redirection, surveillance, and censorship. To address these issues, large organizations such as Google and Cloudflare are deploying standardized protocols that encrypt DNS traffic between end users and recursive resolvers: DNS-over-TLS (DoT) and DNS-over-HTTPS (DoH). In this paper, we examine whether encrypting DNS traffic can protect users from traffic analysis-based monitoring and censoring. We propose a novel feature set to perform traffic analysis attacks, as the features used to attack HTTPS or Tor traffic are not suitable for DNS' characteristics. We show that traffic analysis enables the identification of domains with high accuracy in closed and open world settings, using 124 times less data than attacks on HTTPS flows. We also show that DNS-based censorship is still possible on encrypted DNS traffic. We find that factors such as end-user location, recursive resolver, platform, or DNS client do negatively affect the attacks' performance, but they are far from completely stopping them. We demonstrate that the standardized padding schemes are not effective. Yet, Tor -which does not effectively mitigate traffic analysis attacks on web traffic- is a good defense against DoH traffic analysis.