A mathematical surface is a two-dimensional shape that exists in three-dimensional space. It can be defined by a mathematical equation or a set of equations, which describe how points on the surface relate to each other. Common examples of mathematical surfaces include planes, spheres, and cylinders, each having unique properties and equations. Surfaces can be classified into various types, such as smooth surfaces, which have no sharp edges, and singular surfaces, which may have points where the surface is not well-defined. Understanding these surfaces is essential in fields like geometry, calculus, and computer graphics.