Mathematical theories are systematic frameworks that explain and predict mathematical phenomena. They consist of definitions, axioms, and theorems that provide a structured way to understand relationships and patterns in numbers, shapes, and other mathematical objects. Examples include number theory, which studies properties of integers, and geometry, which focuses on the properties of space and figures. These theories are built upon logical reasoning and proofs, ensuring that conclusions drawn are valid and reliable. They serve as the foundation for various branches of mathematics, influencing fields such as physics, computer science, and economics, where mathematical principles are applied to solve real-world problems.