Combinatory logic is a branch of mathematical logic that focuses on the combination of functions without the need for variables. It simplifies expressions by using combinators, which are higher-order functions that can take other functions as inputs and produce new functions as outputs. This approach is significant in the study of computation and programming languages. Developed by Moses Schönfinkel and later expanded by Haskell Curry, combinatory logic serves as a foundation for functional programming. It emphasizes the application of functions to arguments, allowing for the construction of complex operations from simpler ones, thus providing a framework for understanding computation and reasoning about functions.