Computability theory is a branch of mathematical logic that studies what problems can be solved by algorithms. It explores the limits of computation, determining which functions can be computed and which cannot. Key concepts include Turing machines, recursive functions, and decidability, which help classify problems based on their solvability. One of the central questions in computability theory is whether a given problem can be solved by a computer in a finite amount of time. This leads to the distinction between computable and non-computable functions, shaping our understanding of what computers can and cannot do.