The Leray-Schauder degree is a topological tool used in mathematics, particularly in the field of functional analysis. It helps to determine the number of solutions to a certain type of equation, specifically nonlinear equations. This degree is an extension of the concept of degree of a mapping, which counts the number of times a function wraps around a point. In practical terms, the Leray-Schauder degree provides a way to analyze the behavior of continuous functions on a compact domain. It is particularly useful in proving the existence of solutions to differential equations and other mathematical problems, offering insights into the structure of solution sets.