Summary: We consider a general class of multiprocessor shop scheduling problems, preemptive or non-preemptive, with precedence constraints between operations, with job or operation release dates, and with a class of objective functions including weighted sums of job, operations and stage completion times. We present a general approximation method combining a linear programming relaxation in the operation completion times, with any algorithm for the makespan version of these problems without release dates. If the latter produces a schedule with makespan no larger than \(\rho\) times the trivial lower bound consisting of the largest of all stage average loads (or congestion) and job lengths (or dilation), then our method produces a feasible schedule with minsum objective no larger than \(2e\rho\) times the optimum where \(2e\approx 5.44\). This leads, in particular, to a polynomial time algorithm with polylogarithmic performance guarantee for the minsum multiprocessor dag-shop problem... Our results extend to a broad class of minsum objective functions including some convex objectives related to load balancing. We then present an improved 5.83-approximation algorithm for the open shop problem with total weighted job completion time objective. We conclude with a very simple method which yields O\((m)\)-approximation algorithms for various job shop problems (preemptive, non-preemptive, and no-wait) with m single-processor stages and total weighted job completion time objective.