In this talk, we establish local smoothing estimates for a class of third-order pseudo-differential operators with variable coefficients, which, in particular, includes operators of KdV type. After introducing the main structural properties of these operators, we will outline the key ingredients of the proof, namely Doi’s lemma and suitable energy estimates. We will then present a local well-posedness result that follows from the smoothing effect. This talk is based on joint work with D. Tramontana.