Sometimes things move; sometimes they stay in the same place. These are old, simple and primitive thoughts. Even so, they pervade the most subtle and sophisticated modern physical theories. I will discuss the impact of relativity theory on these old simple ideas. I hope to highlight some simple, basic presuppositions in classical thinking about rest and motion that have turned out false and which are still widely thought to be obvious. I have to say a little about coordinate systems for spacetimes or frames of reference corresponding to them. But I look at them only to draw some lessons that might be applied to the older, simpler ideas. I try to give a rational reconstruction of our simplest ideas of rest and motion; so I offer speculation rather than proof. Though I am guessing, I hope the guess is educated enough to shed some light and hold some interest.