Brouwer fixed-point theorem on non-convex sets

Question

I read about the Brouwer fixed-point theorem in Wikipedia, and got confused about whether it holds when the domain of the function is non convex. On one hand, convexity is explicitly mentioned as a pre-condition. On the other hand, they say that it holds also for domains that are homeomorphic to a disc, and there may be such domains which are not convex (e.g. a concave quadrangle). So, what is the largest family of sets of $\mathbb{R}^n$ on which the BFPT holds?