Our subjective experience would have it that time "moves forward": the past is no longer, and the future is indeterminate and "hasn't happened yet." But it can't actually work that way: special relativity tells us that there's no absolute space of simultaneity; given two spacelike separated events, whether one happened "before" or "after" the other depends on where you are and how fast you're going. This leads us to a "block universe" view: our 3+1 dimensional universe, past, present, and future, simply exists, and the subjective arrow of time somehow arises from our perspective embedded within it.
Without knowing much in the way of physics or cognitive science myself, I can only wonder if there aren't still more confusions to dissolved, intuitions to be unlearned in the service of a more accurate understanding. We know things about the past from our memories and by observing documents; we might then say that memories and documents are forms of probabilistic evidence about another point in spacetime. But predictions about the future are also a form of probabilistic evidence about another point in spacetime. There's a sort of symmetry there, isn't there? Could we perhaps imagine that minds constructed differently from our own wouldn't perceive the same kind of arrow of time that we do?