The area co-ordinate method has been successfnlly applied to constrllct triangular elements. In this paper, this method is generalized to construct quadrilateral elements, and the area co-ordinate theory for quadrilateral elements is systematically developed: (i) the area co-ordinates (L1, L2, L3, L4) of any point in a quadrilateral are defined and two identical equations, which the fonr area co-ordinates should satisfy, are formnlated and proved; (ii) two characteristic parameters for quadrilateral elements are defined and the degeneration conditions under which a qnadrilateral degenerates into a parallelogram or a trapezoid or a triangle are given; (iii) transformation relations between the area co-ordinates and the Cartesian or isoparametric co-ordinates are presented, and several important formnlae for quadrilateral area co-orchnates are listed. Copyright