the classical cauchy integral is a fundamental object of complex analysis whose analytic properties are intimately related to the geometric properties of its supporting curve. in this talk i will begin by reviewing the most relevant features of the classical cauchy integral. i will then move on to the (surprisingly more involved) construction of the cauchy integral for a hypersurface in $\mathbb c^n$. i will conclude by presenting new results joint with e. m. stein concerning the regularity properties of this integral and their relations with the geometry of the hypersurface. (time permitting) i will discuss applications of these results to the szeg\h o and bergman projections (that is, the orthogonal projections of the lebesgue space $l^2$ onto, respectively, the hardy and bergman spaces of holomorphic functions).