ABSTRACT: The program of A. Calderon to study the Cauchy integral along Lipschitz curves in the early sixties has brought into existence a family of operators called the bilinear Hilbert transforms. The study of these operators has proved to be so difficult and involved, that only recently some of their most important properties were discovered. The study of these operators heavily relies on delicate orthogonality and combinatorial arguments and has revealed remarkable and unexpected connections with one of the most challenging problems in analysis, the almost everywhere convergence of Fourier series.