Utility and usability of projective resolutions

Projective resolutions are a standard tool of homological algebra that allow to compute cohomology and associated invariants such as Betti numbers which are used in both abstract contexts and concrete applications. From a theoretical perspective, all projective resolutions are (homotopy) equivalent. Projective resolutions can appear in a wide range of flavours, some more apt for abstract arguments, others more apt for concrete calculations. It is often a change of perspective on the object one is trying to resolve that can make an untractable problem suddenly tractable which I will illustrate in several examples.