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Computability is a fundamental concept in theoretical computer science and mathematical logic that refers to the ability of a problem to be solved by a computational model, such as a Turing machine. It is closely related to the study of algorithms and their capabilities.

The notion of computability was first formally addressed by mathematicians in the 1930s, notably by Alan Turing with his concept of the Turing machine, and by Alonzo Church with his lambda calculus. These models provided a precise definition of what it means for a function or problem to be computable.

A problem is considered computable if there exists an algorithm that can solve it in a finite number of steps. This means that for any given input, the algorithm will halt and produce a correct output. The set of computable problems is also known as the set of decidable problems.

However, not all problems are computable. Some problems are undecidable, meaning there is no algorithm that can solve them for all possible inputs. A classic example is the halting problem, which asks whether a given program will finish running or continue to run forever. Alan Turing proved that the halting problem is undecidable.

The study of computability also involves understanding the limitations of computational models. For instance, while Turing machines and lambda calculus are equivalent in terms of computability, other models like finite automata are less powerful and can only compute a subset of computable functions.

In summary, computability theory provides a framework for understanding the capabilities and limitations of computation, and it plays a crucial role in the development of computer science and mathematics.