This chapter discusses optimization problems in the cone of positive semidefinite matrices, and the duality theory for such ‘linear’ problems. We relate convex rotationally invariant matrix functions to convex functions of the spectrum; this allows us to compute the conjugate of the logarithmic barrier function and the dual of associate optimization problems. The semidefinite relaxation of problems with nonconvex quadratic cost and constraints is presented. Second-order cone optimization is shown to be a subclass of semidefinite programming. The second part of the chapter is devoted to semi-infinite programming and its dual in the space of measures with finite support, with application to Chebyshev approximation and to one-dimensional polynomial optimization.