We study single-machine scheduling problems with step-learning to minimize the total weighted late work. Step-learning is a mechanism that allows jobs to be completed more efficiently if started after a job-dependent learning-date. We show that this problem is strongly NP -hard when both learning-dates and due-dates are arbitrary, but when at least one is assumed to be the same for all jobs, the problem is NP -hard in the ordinary sense, and we provide pseudo-polynomial algorithms for these cases.