When we learn something new, our brain may store the information in synapses or in reverberating loops of electrical activity, but current theories of motor learning focus almost entirely on the synapses. Here we show that loops could also play a role and would bring advantages: loop-based algorithms can learn complex control tasks faster, with exponentially fewer neurons, and avoid the problem of weight transport. They do all this at a cost: in the presence of long feedback delays, loop algorithms cannot control very fast movements, but in this case, loop and synaptic mechanisms can complement each other—mixed systems quickly learn to make accurate but not very fast motions and then gradually speed up. Loop algorithms explain aspects of consolidation, the role of attention, and the relapses that are sometimes seen after a task has apparently been learned, and they make further predictions.