The problem is discussed of whether a traveler can reach a remote object and return sooner than a photon would when taking into account that the traveler can partly control the geometry of his world. It is argued that under some reasonable assumptions in globally hyperbolic space-times the traveler cannot hasten reaching the destination. Nevertheless, it is perhaps possible for the traveler to make an arbitrarily long round-trip within an arbitrarily short (from the point of view of a terrestrial observer) time.

We define the time travel paradox in physical terms and prove its existence by constructing an explicit example. We argue further that in theories -- such as general relativity -- where the spacetime geometry is subject to nothing but differential equations and initial data no paradoxes arise.

In a number of papers it has been claimed that the time machine are quantum unstable, which manifests itself in the divergence of the vacuum expectation value of the stress-energy tensor $\langle{\bf T}\rangle$ near the Cauchy horizon. The expression for $\langle{\bf T}\rangle$ was found in these papers on the basis of some specific approach \cite{Fro,KimT}.\par We show that this approach is untenable in that the above expression firstly is not derived from some more fundamental and undeniable premises, as it is claimed, but rather postulated and secondly contains undefined terms, so that one can neither use nor check it. As a counterexample we cite a few cases of (two-dimensional) spacetimes containing time machines with $\langle{\bf T}\rangle$ bounded near the Cauchy horizon.Comment: LaTex, 12 pages, no figure

A class of static Lorentzian wormholes with arbitrarily wide throats is presented in which the source of the weak energy condition violations required by the Einstein equations is the vacuum stress-energy tensor of the neutrino, electromagnetic, or massless scalar field.

New, more restrictive bounds on photon-massless-pseudoscalar coupling are discussed.