This paper presents a mathematical digital surface definition. This definition is able to deal with boundary points as well as `inner' points. Namely, it is able to distinguish inner points from boundary points. The definition is intuitive and provides a basis for designing fast algorithms for surface decision, boundary search, and surface tracking problems. This definition is equivalent to Morgenthaler and Rosenfeld's definition on simple closed digital surfaces. The definition can be extended to the problem of defining a digital manifold.