Fast Forwarding is an option that allows the player to make the game go a little faster and turning it back to normal, added in v4.4. It is activated by pressing it while in a level. It makes the waves and your Pokemon attacks speed up. In v4.5 , the possible speed options changed from x1 to x2, to x1 to x4.
Added in v4.5, when a shiny or legendary Pokemon appeared on screen, the game would return to normal (x1) speed (this was determined by the fans through a poll on Sam's blog). At the moment, legendary Pokemon cannot be caught (except Moltres, Articuno and Zapdos), but a screen will appear when a shiny pokemon appears.
In v4.6, the same will happen with achievement Pokemon.
The Fast Forwarding button was updated in v4.9 so that you could select a speed without going through the others first.
In v4.9.5, you can now see which speed you are at with a yellow circle around it.
The Fast Forward button was reused in PTD2. It was seen in the topright corner of the screen.