The Cauchy opertaor is one of the most fascinating mathematical constructs, interfacing tightly with classical Harmonic Analysis, Spectral Theory, and Partial Differential Equations. In this talk, which is based on joint work with Irina Mitrea (Temple University, PA, USA) and Michael Taylor (University of North Caroline at Chapel Hill, NC, USA), I will explore the extent to which one may employ complex analysis techniques in purely real variable settings, inspired by the theory built around the classical Cauchy integral operator.