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We analyze the optimal portfolio policy for a multiperiod mean-variance investor facing multiple risky assets in the presence of general transaction costs. For proportional transaction costs, we give a closed form expression for a no-trade region, shaped as a multi-dimensional parallelogram, and show how the optimal portfolio policy can be efficiently computed for many risky assets by solving a single quadratic program. For market impact costs, we show that at each period it is optimal to trade to the boundary of a state-dependent rebalancing region. Finally, we show empirically that the losses associated with ignoring transaction costs and behaving myopically may be large. (C) 2016 Elsevier B.V. All rights reserved.