Given a CW complex X equipped with a G-action, one may define the space of homotopy fixed points with a canonical map to the ordinary fixed points. How much information about the ordinary fixed points can we extract from the homotopy fixed points? Elmendorf's theorem gives us a reason that we should expect little. However, under the assumption that G is a finite p-group, much more can be said through Smith theory and the Sullivan conjecture. In this talk, we will explain these ideas in more detail, as well as the coarse and fine models of equivariant homotopy theory lurking in the background.