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Circular matrices are one of the best studied ensembles in random matrix theory. This talk focuses on CLTs concerning linear statistics of such ensembles. Unlike in classical probability, eigenvalues of random matrices are highly correlated which necessarily requires a different, more subtle, approach to the CLT. In this talk we review the relevant literature and tools such as the moment method or the Stieltjes transform. I will also describe some recent results and future research plans.