Mozilla uses Rust as the language for writing Servo, its new web browser engine designed for parallelism and safety (https://servo.org/).
Due to Rust's compiler design, many kinds of browser security bugs are prevented automatically. In 2013, Samsung got involved, porting Servo to Android and ARM processors. Servo is itself an open source project with more than 750 contributors. It is under heavy development, and amongst other parts it already has its own CSS3 and HTML5 parser implemented in Rust. It passed the web compatibility browser test ACID2 in March 2014 (http://en.wikipedia.org/wiki/Acid2/).
Servo currently supports Linux, OS X, Windows, and Android. Parts of Servo are merged into Gecko (the engine on which Firefox is based), thus lending the Servo project's advancements to Firefox.