Conditions for existence and stability of stationary periodic solutions of uncertain periodic systems are addressed in this paper. Existence and harmonic performance are studied using integral quadratic constraints (IQCs) defined on the space of square integrable periodic functions. Stability is investigated using IQCs defined on the usual space of square integrable functions. The analysis results in criteria formulated as affinely parameterized operator inequalities. Methods of convex optimization are used to find feasible parameters. Two applications will be discussed. The first is harmonic analysis of an electronic circuit and the second involves estimation of the amplitude of a periodic disturbance in a nonlinear control system.