An important question in algorithmic solution is whether a given problem’s solvability is decidable or not. The problems of decidable type have answer in the form of Yes/No, while for the undecidable type, no algorithm can exist for their solution. This chapter takes us from foundational concepts to theories about this question, and introduces with what problems are solvable (decidable)/semi-decidable/undecidable, how to formulate them, and presents the characteristics of an effective procedure. In addition, we shall arrive to relation between decision problems vs. recursive languages, what are the decidable and closure properties of formal languages. We will see how the recursive languages differ from recursively enumerable, what are the properties of computable, non-computable, countable, and uncountable sets. The undecidability theory starts with Hilbert—one of his problems seeks to answer about an arbitrary diophantine equation with rational integral coefficients, whether it has a solution in rational integrals. As per Gödel, the symbolic logic problems can be transformed into positive integers in recursive set. Therefore, whether a problem is solvable or not, amounts to say whether the problem of positive integers is solvable or not in the set of integers. The chapter presents how to represent a formula into a number, Gödel’s Incomplete theorem, and the unsolvability has been illustrated through examples, particularly through diagonal language, Barber’s paradox, and through undecidability of halting problem. Many undecidability results have been proved through Rice’s theorem. The chapter concludes with its summary, self review questions and exercises.