Let Gamma_(x,y)=(t, g(x,y,t)) be a variable curve in the plane, where here  t is in R and (x,y) is in R² while

g_(x,y)(-) := g(x,y,-) : R --> R

is a "suitable" real function. Under what conditions on the curve Gamma_(x,y) - (our main target: minimal regularity in x and y) - do we have that the Hilbert transform along curve Gamma defined by

H_{Gamma}f(x,y):=p.v.\,\int_{\R} f(x-t,\,y+\g(x,y,t))\,\frac{dt}{t}

is a bounded operator from L^p(R^2) to L^p(R^2) for 1<p<\infty? We will insist on the history and motivation for this problem from both 

* the PDE side - the study of constant coefficient parabolic differential operators and 
* the Harmonic Analysis side - connections with singular maximal operators and the conjecture of A. Zygmund on maximal integrals along (Lipschitz) vector fields.

If the time allows we will very briefly outline some of the methods we used in obtaining our most recent results.