Theoretical physicists have not wanted for imagination when it comes to developing a quantum theory of gravity. String theory, loop quantum gravity, causal-set theory, twistor theory: the approaches are diverse and the disagreements among their proponents are often vehement. And yet they have a common feature: that classical spacetime is not a fundamental ingredient of the world, but a construction consisting of more fundamental degrees of freedom. Those degrees of freedom become structured in very specific ways to give rise to the observed features of classical spacetime. I'll discuss the interpretational implications of several leading theories.