A few of them. It won't stop logic errors (although enums help here) but it will prevent buffer overflows, use-after-free, some cases of type confusion, uninitialised reads, and a few other problems.

If you look at the CVEs for browsers, I would say a great majority is due to errors that would've been prevented in any memory safe language - not specifically Rust.

However due to the other requirements (like performance) and its adoption at Mozilla, Rust is a real contender.