Why are integral group rings so important in topology?

Question

Having worked on group rings $\mathbb{Z}[G]$ for the last year, I am beginning to feel quite comfortable with them. I also know of a few applications to topology. However, these are individual uses whereas integral group rings seem to be fundamental in some sense in algebraic topology (they keep cropping up!). So I was wondering what makes them so crucial? What is it about them that makes them more important than, say, $R[G]$ ($R$ commutative). Is this also a feature shared in geometry, i.e. is $\mathbb{Z}$ just as important in the realms of algebraic geometry?