  In this paper we deal with the existence of periodic orbits of geodesible
vector fields on closed 3-manifolds. A vector field is geodesible if there
exists a Riemannian metric on the ambient manifold making its orbits geodesics.
In particular, Reeb vector fields and vector fields that admit a global section
are geodesible. We will classify the closed 3-manifolds that admit aperiodic
volume preserving real analytic geodesible vector fields, and prove the
existence of periodic orbits for real analytic geodesible vector fields (not
volume preserving), when the 3-manifold is not a torus bundle over the circle.
We will also prove the existence of periodic orbits of C2 geodesible vector
fields in some closed 3-manifolds.
