Mathematical theories are systematic frameworks that explain and predict mathematical phenomena. They consist of definitions, axioms, and theorems that help mathematicians understand relationships and structures within mathematics. Examples include number theory, which studies integers, and geometry, which focuses on shapes and their properties. These theories are built upon logical reasoning and proofs, ensuring that conclusions drawn are valid. They often lead to new discoveries and applications in various fields, such as physics, computer science, and economics. By providing a foundation for problem-solving, mathematical theories play a crucial role in advancing knowledge and technology.