  The bumblebee field coupled with gravity is considered. This gravitational
theory exhibits spontaneous breaking of Lorentz symmetry. The G\"{o}del-type
universe is introduced and then the causality and its violation are studied.
Causal and non-causal G\"{o}del-type solutions are obtained for different
content of matter. In addition, when the coupling constant, which controls the
interaction between the bumblebee field and the gravitational field, is zero
the bumblebee potential may be associated with the value of the cosmological
constant of the general relativity. Furthermore, the case with the non-zero
coupling constant is also investigated.
