We investigate the control of constrained stochastic linear systems when faced with only limited information regarding the disturbance process, i.e. when only the first two moments of the disturbance distribution are known. We consider two types of distributionally robust constraints. The constraints of the first type are required to hold with a given probability for all disturbance distributions sharing the known moments. These constraints are commonly referred to as distributionally robust chance constraints with second-order moment specifications. In the second case, we impose conditional value-at-risk (CVaR) constraints to bound the expected constraint violation for all disturbance distributions consistent with the given moment information. Such constraints are referred to as distributionally robust CVaR constraints with second-order moment specifications. We argue that the design of linear controllers for systems with such constraints is both computationally tractable and practically meaningful for both finite and infinite horizon problems. The proposed methods are illustrated for a wind turbine blade control design case study where flexibility issues play an important role, and for which distributionally robust constraints constitute sensible design objectives.