The study of fixed points of group actions is an important topic
in geometry and topology. In this talk, we focus on fixed points of
actions in the case where manifolds admit symplectic structures and circle
actions on the manifolds preserve the symplectic structures. We discuss main
theorems on fixed points of symplectic circle actions, and their
relation to the question of when symplectic actions are Hamiltonian. 
Next, we study properties of symplectic circle actions when the
fixed points are isolated, and discuss the classification of symplectic
circle actions, when the number of fixed points is small.