Some of the issues below are commonly recognized as problems per se, i.e., it is general agreement that the solution is unknown. Others may be described as controversies, i.e., while there is no common agreement about the answer, there are established schools of thought that believe they have a correct answer.
Is the human ability to use syntax based on innate mental structures or is syntactic speech the function of intelligence and interaction with other humans? The question is tightly related with the two major problems: language emergence and language acquisition.
The language acquisition device: How localized is language in the brain? Is there a particular area in the brain responsible for the development of language abilities, or is language not localized in the brain, or is it only partially localized?
What fundamental reasons explain why ultimate attainment in second language acquisition is typically some way short of the native speaker's ability, with learners varying widely in performance?
"Virtually every generative linguist has had the following experience: a given linguistic entity (sentence, novel word, pronunciation) is presented to a native speaker and judged to be neither fully well-formed nor fully unacceptable. In such instances, consultants often say things like "I guess I could say that," "It's all right but not perfect," "It's pretty bad but not completely out," and the like"
^ Robert Spence, "A Functional Approach to Translation Studies. New systemic linguistic challenges in empirically informed didactics", 2004, ISBN 3-89825-777-0, thesis. A pdf file
^ Jeffrey C. Reynar, "Topic Segmentation: Algorithms and Applications" (1998), Ph.D thesis, citation.
^ Pierre Isabelle, "Another Look at Nominal Compounds", In Proc. of the 10th International Conference on Computational Linguistics(pdf)