Internet users rely on the availability of websites and digital services to engage in political discussions, report on newsworthy events in real-time, watch videos, etc. However, sometimes those who control networks, such as governments, censor certain websites, block specific applications or throttle encrypted traffic. Understandably, when users are faced with egregious censorship, where certain websites or applications are banned, they seek reliable and efficient means to circumvent such blocks. This tension is evident in countries such as a Iran and China, where the Internet censorship infrastructure is pervasive and continues to increase in scope and effectiveness. An arms race is unfolding with two competing threads of research: (1) network operators' ability to classify traffic and subsequently enforce policies and (2) network users' ability to control how network operators classify their traffic. Our goal is to understand and progress the state-of-the-art for both sides. First, we present novel traffic analysis attacks against encrypted communications. We show that state-of-the-art cryptographic protocols leak private information about users' communications, such as the websites they visit, applications they use, or languages used for communications. Then, we investigate means to mitigate these privacy-compromising attacks. Towards this, we present a toolkit of cryptographic primitives and protocols that simultaneously (1) achieve traditional notions of cryptographic security, and (2) enable users to conceal information about their communications, such as the protocols used or websites visited. We demonstrate the utility of these primitives and protocols in a variety of real-world settings. As a primary use case, we show that these new primitives and protocols protect network communications and bypass policies of state-of-the-art hardware-based and software-based network monitoring devices.