This work unifies and extends many generalizations of Borsuk's theorem. The authors prove an equivariant version of the Kuratowski-Dugundji Theorem. In order to do this, they show the existence of a (quasi)- fundamental domain for any free action of a compact Lie group on a metric space \(X\). Then for \(G\) finite and \(\Psi, \varphi : M \to S\) a pair of equivariant maps where \(M\) is a manifold of the same dimension \(n\) as the sphere \(S\), they show that \(\deg \Psi - \deg \varphi \equiv 0 \pmod {\text{GCD} \{|G/H_i |\}}\) under certain hypotheses. Finally they show that \(n - k\) is a lower bound for the genus of \(S\smallsetminus B\) where \(S\smallsetminus B\) is a free \(G\)-subspace and \(B\) the image of a smooth mapping of a \(k\)-dimensional compact manifold.