I cannot say I am very familiar with ARMA (I must admit that I am kind of biased to begin with, so for a long time, I haven't tried to bother with AR/MA-like linear models).
However, for some reason, I tried ARMA to solve problems of forecasting the behavior of some very complicated dynamic systems, and I found ARMA models work very well, much better than most of the modelling methods I am familiar with.
However, looking at the method, it seems that there is nothing particularly special about these kinds of time series models, like OLS, regression, MLE etc, and it looks like theoretically speaking it is even very limited comparing to nonlinear models.
So what do you think makes AR/MA work better than many more general models? the properties of stationary processes? MLE? The use of some information theory in training?