This article is devoted to the investigation of wrap groups of connected fiber bundles. These groups are constructed with mild conditions on fibers. Their examples are given. It is shown, that these groups exist and for differentiable fibers have the infinite dimensional Lie groups structure, that is, they are continuous or differentiable manifolds and the composition (f, g) ⟼ f−1g is continuous or differentiable depending on a class of smoothness of groups. Moreover, it is demonstrated that in the cases of real, complex, quaternion and octonion manifolds these groups have structures of real, complex, quaternion or octonion manifolds respectively. Nevertheless, it is proved that these groups does not necessarily satisfy the Campbell-Hausdorff formula even locally.