I haven't studied anything related to linguistics since high-school, so I'm not sure if this question makes any sense and is on-topic.
I would like to know if there are open problems related to grammars. What are the main lines of research? I was thinking whether every grammar can analyse correctly any sentence, axiomatic grammars, incompleteness, consistency and things like that.
The question is probably too broad, but I haven't seen any similar question and I think that it would be useful to have some broad questions in the beta phase.
But if the answer to the question is just no, it won't have much value.