In this paper, we examine embeddings and immersions f:M -> E^3, where M is a smooth and polyhedral closed surface. We define the concept of a transversal line, geometric degree of a surface, and algebraic degree of a surface. We then relate the geometric degree, algebraic degree, and Euler characteristic. The concept of tightness for a surface is defined, we this is also related to the degree of a surface. Finally, several generalizations into higher dimensional space are given.