Animals can learn to associate a behavior or a stimulus with a delayed reward, this is essential for survival. A mechanism proposed for bridging this gap are synaptic eligibility traces, which are slowly decaying tags, which can lead to synaptic plasticity if followed by rewards. Recently, experiments have demonstrated the existence of synaptic eligibility traces in diverse neural systems, depending on either neuromodulators or plateau potentials. Evidence for both eligibility trace-dependent potentiation and depression of synaptic efficacies has emerged. We discuss the commonalities and differences of these different results. We show why the existence of both potentiation and depression is important because these opposing forces can lead to a synaptic stopping rule. Without a stopping rule, synapses would saturate at their upper bound thus leading to a loss of selectivity and representational power. We discuss the possible underlying mechanisms of the eligibility traces as well as their functional and theoretical significance.