We define and study a new notion of "robust simulations" between complexity classes which is intermediate between the traditional notions of infinitely-often and almost-everywhere, as well as a corresponding notion of ``significant separations''. A language L has a robust simulation in a complexity class C if there is a language in C which agrees with L on arbitrarily large polynomial stretches of input lengths. There is a significant separation of L from C if there is no robust simulation of L in C. The new notion of simulation is a cleaner and more natural notion of simulation than the infinitely-often notion. We show that various implications in complexity theory such as the collapse of PH if \NP = P and the Karp-Lipton theorem have analogues for robust simulations. We then use these results to prove that most known separations in complexity theory, such as hierarchy theorems, fixed polynomial circuit lower bounds, time-space tradeoffs, and the recent theorem of Williams, can be strengthened to significant separations, though in each case, an almost everywhere separation is unknown. Proving our results requires several new ideas, including a new proof of the nondeterministic-time hierarchy theorem. This is joint work with Rahul Santhanam (Edinburgh) from the recent ICALP conference. The paper is available at http://www.cs.uchicago.edu/~fortnow/papers/robust.pdf