Computability Theory is a branch of mathematical logic that explores what problems can be solved by algorithms. It investigates the limits of computation, focusing on questions like whether a problem can be solved by a computer and how efficiently it can be done. Key concepts include Turing machines, which are abstract devices that help us understand the nature of computation, and decidability, which determines if a problem can be resolved with a yes or no answer. This field also examines complexity classes, which categorize problems based on the resources needed to solve them, such as time and space. By studying these concepts, computability theory helps us understand the capabilities and limitations of computers, guiding the development of algorithms and programming languages.