Fixed point theory studies the extent to which fixed points of a self map of a space are intrinsic. In many mathematical settings, the existence of a solution can be rephrased in terms of the existence of a fixed point for an appropriate map, leading to applications across mathematics. Variations of fixed point problems have solutions which use tools from stable and equivariant homotopy theory. In this talk, I will discuss progress towards a new homotopy theory for spaces with a finite group action and its applications to the long term goal of solving fixed point problems for homeomorphisms of manifolds.