In many species and behaviors, learning and memory formation involve plasticity in at least two distinct neural pathways, responsible for short and long-term learning, and a process of consolidation requiring interaction between them. A well-known example is the consolidation of hippocampal memory traces into the neocortex in mammals. Consolidation mechanisms are also observed in motor learning (between cortical and subcortical structures) and associative learning in insects (between subregions of the mushroom body). Here, we propose a model that captures common computational principles underlying these phenomena. The key component of our model is recall-gated consolidation, in which the long-term pathway prioritizes the storage of memory traces that are familiar to the short-term pathway. This mechanism shields long-term memory from spurious synaptic changes, enabling it to focus on reliable signal in the environment. We show that this model has significant advantages, substantially amplifying the signal-to-noise ratio with which intermittently reinforced memories are stored. In fact, we demonstrate mathematically that these advantages surpass what is achievable by synapse-local mechanisms alone, providing a motivation for systems (as opposed to synaptic) consolidation. We describe neural circuit implementations of our general model for different types of learning problems, which make use of interpretable factors such as prediction accuracy, confidence, or familiarity to modulate the rate of consolidation. Our model makes a number of predictions, including (1) that the rate of memory consolidation should increase with the number and consistency of training repetitions (2) that short-term memory pathways benefit from sparser, higher-dimensional representations than long-term pathways, (3) that recall performance depends non-monotonically on the spacing of training trials, and the optimal training spacing depends on the time scale of evaluation. These predictions are largely consistent with existing evidence from the mammalian and insect memory consolidation literature, while also motivating new experiments to test them directly.