Computability theory studies whether a computational problem can be solved by an algorithm regardless of time or memory constraints. This work introduces the concept of computable and non-computable problems, explains the Halting Problem, and discusses the theoretical model of computation known as the Turing Machine. The study also highlights the classification of problems into decidable and undecidable categories and explains the difference between computability and complexity. The topic forms a fundamental basis of theoretical computer science and has applications in artificial intelligence, cybersecurity, formal logic, and software verification.