Let T be a homeomorphism of odd prime period p on an oriented integral Poincar6 space X 2". The main theorem of this paper is an analogue in this context of the Atiyah-Singer-Segal G-Signature Theorem. It computes the Witt class of the Zp quadratic form arising from the fixed set F in terms of the orthogonal (or symplectic, if n is odd) representation of T* in H"(X2")/Tor. We shall also prove a similar theorem for circle actions. Our proofs owe much to the paper of Bredon [5]. As Bredon has noted, these theorems yield new information even for differentiable actions on certain non-closed manifolds. Before precisely stating our principal results, we need some assumptions and definitions. Blanket Assumptions. All spaces are assumed to be Hausdorff and to have finite covering dimension. In order that (~ech and singular cohomology will agree, all integral Poincar6 spaces, integral Poincar6 pairs, and the doubles of these pairs are assumed to be HLC [6]. 9