Publisher Summary This chapter discusses the rational points on cubic surfaces. No method is known for determining whether rational points exist on a general cubic surface f (x, y, z) = 0, or for finding all of them if any exist. Geometric considerations may help in finding infinite solutions and even all the solutions for cubic surfaces. All the rational points on a cubic surface can be found if it contains two lines whose equations are defined by conjugate numbers of a quadratic field and in particular by rational numbers. Several theorems associated with the rational points on cubic surfaces are also discussed in the chapter.