  The linear stability of closed timelike geodesics (CTGs) is analyzed in two
spacetimes with cylindrical sources, an infinite rotating dust cylinder, and a
cylindrical cloud of static cosmic strings with a central spinning string. We
also study the existence and linear stability of closed timelike curves in
spacetimes that share some common features with the G\"odel universe
(G\"odel-type spacetimes). In this case the existence of CTGs depends on the
`background' metric. The CTGs in a subclass of inhomogeneous stationary
cosmological solutions of the Einstein-Maxwell equations with topology $
S^3\times \mathbb R$ are also examined.
