A complex polynomial defines a holomorphic vector field in the complex plane. The quasi-conformal conjugacy class of the polynomial is completely determined by a combinatorial invariant. Furthermore, within each combinatorial class the polynomial is uniquely determined by a finite number (settled by the combinatorial class) of complex numbers. This fundamental classification of complex polynomial vector fields is proved using surgery. Further developments are to classify possible bifurcations, to understand the decomposition of parameter spaces due to the different combinatorial classes and the bifurcations among them, and also to extend to meromorphic vector fields arising from rational functions on the Riemann sphere.