The Art Gallery Problem is the problem of determining the number of observers necessary to cover an art gallery such that every point is seen by at least one observer. This problem is well known and has a linear time solution for the 2 dimensional case, but little is known about 3-D case. In this thesis, the dominance relationship between vertex guards and point guards is searched and found that a convex polyhedron can be constructed such that it can be covered by some number of point guards which is one third of the number of the vertex guards needed. A new algorithm which tests the visibility of two vertices is constructed for the discrete case. How to compute the visible region of a vertex is shown for the continuous case. Finally, several potential applications of geometric terrain visibility in geographic information systems and coverage problems related with visibility are presented.