Let M be a compact C∞ Riemannian manifold, X a Killing vector field on M, and φt its 1-parameter group of isometries of M. In this, paper, we obtain some basic properties of the set of periodic points of φt. We show that the set of least periods is always finite, and the set P(X, t) of points of M having least period t for the vector field X is a totally geodesic submanifold, with possibly non-empty boundary. Moreover, we show there are at least m geometrically distinct closed geodesic orbits of φt, where m is the number of least periods which are not integral multiples of any other least period.