  A study of symplectic actions of a finite group $G$ on smooth 4-manifolds is
initiated. The central new idea is the use of $G$-equivariant
Seiberg-Witten-Taubes theory in studying the structure of the fixed-point set
of these symmetries. The main result in this paper is a complete description of
the fixed-point set structure (and the action around it) of a symplectic cyclic
action of prime order on a minimal symplectic 4-manifold with $c_1^2=0$.
Comparison of this result with the case of locally linear topological actions
is made. As an application of these considerations, the triviality of many such
actions on a large class of 4-manifolds is established. In particular, we show
the triviality of homologically trivial symplectic symmetries of a $K3$ surface
(in analogy with holomorphic automorphisms). Various examples and comments
illustrating our considerations are also included.
