this thesis contains a study of higher dimensional numerical quadrature, especially two dimensional gregory quadrature. in chapter i, we discuss the problem of approximation and integration, the reasons why numerical quadrature has been developed and review some important numerical quadrature formulas. chapter ii contains some fundamental concepts, including some useful notations on higher dimensions, barycentric coordinates, homogeneous representation of polynomials, the newton-cotes polynomials, and integration and differentiation in barycentric coordinates . in chapter iii we present a generalization of newton- cotes quadrature to higher dimensions over k-simplices. our generalization is based on the properties of the newton-cotes lattice and the newton-cotes polynomials. a rather complete list (up to 13th order) of two dimensional newton-cotes quadrature formulas over triangles as well as some three dimensional newton- cotes quadrature formulas over tetrahedra are given. in chapter iv we apply the concept of the hexagonal kpartition of unity, developed by professor p.o. frederickson, to construct some two dimensional gregory quadrature formulas. the general derivation of an mth order gregory quadrature formula over a plane region with piece-wise linear boundary is given. particularly, the gregory quadrature formulas of the first three orders over some special regions are computed.