  In the Rastall gravity a non-minimal coupling between geometry and matter
fields is considered. Then the usual energy-momentum tensor conservation law is
not valid. Here a Lagrangian formalism is proposed to the Rastall theory of
gravity. The G\"{o}del-type universe is studied in this gravitational model.
Then it is studied whether this theory permits causality violation. The field
equations do not exclude solutions with a breakdown of causality for a perfect
fluid as matter content. In this case, an expression for the critical radius
(beyond which the causality is violated) is determined. In addition, for a
combination between perfect fluid and scalar field as matter content the theory
accommodates causal G\"{o}del-type solution.
