This chapter studies the positive semidefinite matrices, focusing on the inequalities of the matrices of this type. Our goal is to present the fundamental results and to show some often-used techniques. Sect. 7.1 gives the basic properties, Sect. 7.2 treats the Loewner partial order of positive semidefinite matrices, and Sect. 7.3 presents some inequalities for principal submatrices. Sect. 7.4 derives determinant inequalities of Fischer, Hadamard, and Hua, and Sect. 7.5 investigates the Hadamard product of the positive semidefinite matrices.