The five-dimensional ( 5D ) Riemannian Gödel-type manifolds are examined in light of the equivalence problem techniques , as formulated by Cartan . The necessary and sufficient conditions for local homogeneity of these 5D manifolds are derived . The local equivalence of these homogeneous Riemannian manifolds is studied . It is found that they are characterized by two essential parameters m ^ { 2 } and \omega : identical pairs ( m ^ { 2 } , \omega ) correspond to locally equivalent 5D manifolds . An irreducible set of isometrically nonequivalent 5D locally homogeneous Riemannian Gödel-type metrics are exhibited . A classification of these manifolds based on the essential parameters is presented , and the Killing vector fields as well as the corresponding Lie algebra of each class are determined . It is shown that apart from the ( m ^ { 2 } = 4 \omega ^ { 2 } , \omega \not = 0 ) and ( m ^ { 2 } \not = 0 , \omega = 0 ) classes the homogeneous Riemannian Gödel-type manifolds admit a seven-parameter maximal group of isometry ( G _ { 7 } ) . The special class ( m ^ { 2 } = 4 \omega ^ { 2 } , \omega \not = 0 ) and the degenerated Gödel-type class ( m ^ { 2 } \not = 0 , \omega = 0 ) are shown to have a G _ { 9 } as maximal group of motion . The breakdown of causality in these classes of homogeneous Gödel-type manifolds are also examined .