I am writing this thread because I have too often had to explain this and I basically wanted to have a resource available to direct folks to, and, frankly, the Wikipedia topics on this are pretty vague on the explanation. Any questions, comments, or corrections are greatly appreciated. So, here we go... Often, the argument is made that the expansion of the universe must be decelerating which is incorrect. The first argument is that more distant galaxies are receding more quickly than nearer galaxies, while the second is that more distant galaxies are older than nearer galaxies. While both statements are scientifically accepted, the conclusion that galaxies were recessing faster in the past and therefore the expansion of the universe is decelerating is completely incorrect. Another incorrect assumption often made is that this expansion, which leads to the recession of distant galaxies, is equated with proper relative motion, and that the galaxies have a relative velocity component due to this recession. This also is not correct. It is more appropriate to consider that galaxies remain fixed, with the empty space between expanding over time. Certainly, there can be true relative motion among galaxies, and it would be quite unphysical if there were not. Before you can convince yourself or anyone else that the expansion of the universe is, in fact, accelerating, you first must understand how the universe would appear to a fixed observer if the expansion rate were constant. I will attempt to explain this as clearly as possible. The above diagram illustrates how this constant rate expansion would occur in a single spatial dimension for seven uniformly spaced galaxies over two time intervals, with time progressing from bottom to top. The green dots represent a fixed observer galaxy, while the blue, purple, and red dots represent progressively more distant galaxies. So initially the blue galaxies are 1 unit away, the purple galaxies are 2 units away, and the red galaxies are 3 units away. After one time interval, the configuration of the galaxies is now such that the blue galaxies are 2 units away, the purple galaxies are 4 units away, and the red galaxies are 6 units away, so space has expanded uniformly to twice it's original size. But notice that the change in distance from the fixed, green observer is progressive: the blue galaxies have recessed by 1 unit, the purple by 2, and the red by 3! Now it should be clear that this change in distance is directly proportional to the observed distance, which in this case, the ratio is 1/2. As mentioned above, this change of distance is entirely different from physical motion, and it should be clear from the diagram that it is happening very uniformly at any instant in time. Space is not expanding faster anywhere regardless of distance. The fact that distant galaxies appear to be receding faster is entirely an observer effect due to the fact that there is more space between an observer and a more distant galaxy. Finally, after a second interval of time, we can see in the diagram that the space between all galaxies has again doubled uniformly, so this is how constant expansion would appear to our fixed observer over time. The blue galaxy is now 4 units away, the purple 8, and the red 12. But again notice that the change in distance is progressive, directly proportional to the observed distance, and that the ratio is still 1/2, which demonstrates that, at least in this scenario, the rate of expansion is constant. So now that we understand the expansion of the universe more clearly, how can we determine if the rate of expansion is decreasing, constant, or, increasing? Well, we can take the top line of the diagram and either slightly stretch it (accelerating expansion) or slightly compress it (decelerating expansion). Now I'll leave it to you to deduce the observational effects of either case, but remember while doing so, that we can only observe the present (galaxies in the top line of the diagram). Hint: red-shift is cumulative over time. As a final note, scientists used this same process to search for evidence that this expansion was slowing down due to gravitational effects. They made specific predictions so they would know where to search, and their observations actually demonstrated the opposite of what was expected. Granted there is still a lot of debate over the accuracy of the measurements and variety of other factors involved in these experiments, but the tide seems to be in favor of accelerating expansion. Sorry Dark Energy Haters.