Harmonic functions are mathematical functions that satisfy Laplace's equation, which states that the second derivatives of the function sum to zero. These functions are smooth and continuous, meaning they have no abrupt changes or discontinuities. They are often used in various fields, including physics and engineering, to model phenomena such as heat distribution and fluid flow. In two or more dimensions, harmonic functions exhibit interesting properties, such as the mean value property, which states that the value of the function at a point is the average of its values over any surrounding sphere. They are closely related to potential theory and are essential in the study of partial differential equations.