Given a single machine and a set of jobs with due dates, the classical NP-hard problem of scheduling to minimize total tardiness is a well-understood one. Lawler gave an FPTAS for it some twenty years ago. If the jobs have positive weights the problem of minimizing total weighted tardiness seems to be considerably more intricate. To our knowledge there are no approximability results for it. In this paper, we initiate the study of approximation algorithms for the problem. We examine first the weighted problem with a fixed number of due dates and we design a pseudopolynomial algorithm for it. We show how to transform the pseudopolynomial algorithm to an FPTAS for the case where the weights are polynomially bounded. For the general case with an arbitrary number of due dates, we provide a quasipolynomial randomized algorithm which produces a schedule whose expected value has an additive error proportional to the weighted sum of the due dates.