What are applications of the levels of the Chomsky hierarchy? I've recently finished a course on languages in computer science, so we covered characteristics of regular, context-free, decidable and semi-decidable languages. While that is all well and good, we never went into any detail on what applications of this knowledge are (other than compilers being context-free language recognizers). I've tried googling for answers but wasn't able to find anything relevant. I'm wondering if someone can just explain what current applications for each 'type' of language are and perhaps link further material for reading. 

Thank you.
First, let's be clear, there are two separate types of languages, Turing Decidable (also called recursive), and Turing Recognizable (also called semi-decidable or recursively-enumerable).

The "application" of decidable languages is all of computer science. It's a huge class that includes exactly the problems that can be solved by algorithms. So you're not going to find specific applications like you would for compilers and CFGs, because the applications covered by decidable languages are "everything else." If you can write code for it, it's in this class.

Recursively-enumerable problems are ones where you know a solution when you see one, but might never know when you're out of potential solutions. You always halt on a "yes" answer, but might never halt for "no". This class is basically first-order theorem proving: we recognize proofs when we see them, but with quantifiers, there's no way to know if no proof exists.

It's also worth mentioning that parsers are CFG recognizers. Other parts of compilers, like translation and type-checking, are definitely not context-free.