Mathematical surfaces are two-dimensional shapes that exist in three-dimensional space. They can be defined by equations or parametric representations, allowing mathematicians to study their properties and behaviors. Common examples include planes, spheres, and more complex shapes like toruses and hyperboloids. These surfaces can be visualized using graphing software or 3D modeling tools, making it easier to understand their structure. Mathematical surfaces are essential in various fields, including geometry, physics, and computer graphics, as they help describe real-world objects and phenomena in a mathematical framework.