In this thesis, we consider the Hermite-Hadamard inequality for convex functions of two variables (HH2). In Chapter 1, a brief introduction of HH2 is given. In Chapter 2, we give two inequalities related to HH2, which are inspired by a result established recently. We apply them to two convex functions. Some results related to the p−logarithmic mean and the arithmetic mean are obtained. Applications of the classical Hermite-Hadamard inequality (for functions of one variable) to matrix functions are also given in this chapter. The thesis ends with a chapter of a brief conclusion.