Comment: This is the final version of the paper accepted for publication at Econometric Theory. Please do not cite earlier versions as they contain errors that were corrected in the review process. I am grateful to Benedikt P\"otscher and to three anonymous referees for the substantial improvements to the paper since it first appeared. Any remaining errors are my own responsibilityThis paper considers linear rational expectations models in the frequency domain. The paper characterizes existence and uniqueness of solutions to particular as well as generic systems. The set of all solutions to a given system is shown to be a finite dimensional affine space in the frequency domain. It is demonstrated that solutions can be discontinuous with respect to the parameters of the models in the context of non-uniqueness, invalidating mainstream frequentist and Bayesian methods. The ill-posedness of the problem motivates regularized solutions with theoretically guaranteed uniqueness, continuity, and even differentiability properties.