Both the gauge groups and 5-manifolds are important in physics and mathematics.In this paper,we combine them to study the homotopy aspects of gauge groups over 5-manifolds.For principal bundles over non-simply connected oriented closed 5-manifolds of a certain type,we prove various homotopy decompositions of their gauge groups according to different geometric structures on the manifolds,and give the partial solution to the classification of the gauge groups.As applications,we estimate the homotopy exponents of their gauge groups,and show periodicity results of the homotopy groups of gauge groups analogous to the Bott periodicity.Our treatments here are also very effective for rational gauge groups in the general context,and applicable for higher dimensional manifolds.