Following Wallace [15], we define an act to be a continuous function ,u: S x X X such that (i) S is a topological semigroup, (ii) X is a topological space, and (iii) ,u(s, ,u(t, x)) =,(st, r) for all s, t E S and x E X. We call (S, X, ,) an action triple, X the state space of the act, and we say S acts on X. We assume all spaces are Hausdorff and write sx for ,u(s, x). S is said to act transitively if Sx= X for all x E X and effectively if sx = tx for all x E X implies that s = t. The first section of this paper deals with transitive actions and especially with the case where the semigroup is simple. We obtain as a corollary that if S is a compact coninected semigroup acting transitively and effectively on a space X that contains a cut point, then K, the minimal ideal of S, is a left zero semigroup and X is homeomorphic to K.. A C-set is a subset, Y, of X with the property that if M is any continuum contained in X with M rc Y# 0, then either Mc Y or Yc M. In the second section, we consider the position of C-sets in the state space and prove as a corollary that if S is a compact connected semigroup with identity acting effectively on the metric indecomposable continuum, X, such that SX= X, then S must be a group. The author wishes to thank Professor L. W. Anderson for his patient advice and criticism.