<font face="Arial, Helvetica, sans-serif" size="2">Quantum impurity problems are among the few nontrivial problems describing strongly correlated condensed matter systems that are amenable to exact solutions. Due to recent advances in nanofabrication techniques, predictions from the conformal field theory (CFT) approach, such as nontrivial scaling exponents, can be tested by quantum dot experiments. A major challenge on which my talk focuses is the extension of exact methods to the non-equilibrium regime, in which quantum dot experiments are typically performed. Using CFT methods we solved exactly the problem of non-equilibrium transport through double quantum dots, obtaining exact results of experimental relevance for the full counting statistics of charge transport near a non-Fermi liquid critical point. Those results also imply a nontrivial effective charge of 2e, which can be probed by shot noise experiments. 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