On the importance of the i.i.d. assumption in statistical learning

I think we can all agree that this assumption is usually violated in practice (think temporal auto-correlation for instance, as observed when dealing with time series). My question is therefore: where exactly does the i.i.d. I'm asking this because I can think of many models (e.g. Actually the results usually stay the same, it is the inferences that one can draw that change (e.g. My guess is therefore that the i.i.d.