I suspect the greatest contribution of the Roman empire was given to us twice over. It was the idea of a greater state, over and above all the petty local divisions.

This vision of power (not original to Rome, see the Persians for the original copy) has gone through the ages and the minds of generations of warlords, kings and politicians and truly shaped our world.

On a second run, the Christian church used the same idea of an overarching power, via the papacy to unite disparate powers. This idea has led to scientific peer review and the united nations. And organised religion I suppose, but I'm not inclined to like that much myself.