Let $v$ be a smooth vector field on the plane, that is a map from the plane to the unit circle. The authors study sufficient conditions for the boundedness of the Hilbert transform $\textrm{H}_{v, \epsilon }f(x) := \text{p.v.}\int_{-\epsilon}^{\epsilon} f(x-yv(x))\;\frac{dy}y$ where $\epsilon$ is a suitably chosen parameter, determined by the smoothness properties of the vector field. Table of Contents: Overview of principal results; Besicovitch set and Carleson's theorem; The Lipschitz Kakeya maximal function; The $L^2$ estimate; Almost orthogonality between annuli. (MEMO/205/965)
