  In this paper the causality issues are discussed in a non-Riemannian
geometry, called Lyra geometry. It is a non-Riemannian geometry originated from
Weyl geometry. In order to compare this geometry with the Riemannian geometry,
the Einstein field equations are considered. It is verified that the G\"{o}del
and G\"{o}del-type metric are consistent with this non-Riemannian geometry. A
non-trivial solution for G\"{o}del universe in the absence of matter sources is
determined without analogue in general relativity. Different sources are
considered and then different conditions for causal and non-causal solutions
are discussed.
