For a manifold M, let S(M) be the set of all manifolds  homotopy equivalent to M. The surgery theory is an effective way of  computing S(M) and answering related homotopy problems. The theory has  four-fold periodicity and functoriality property. I will discuss the  extension of such properties to the equivariant case, i.e., there is a  group acting on M and all the homotopies preserve group actions. I  will also discuss the replacement problem, which studies to what  extent the fixed points of group actions can be homotopically modified.