  This article is devoted to the investigation of wrap groups of connected
fiber bundles over the fields of real $\bf R$, complex $\bf C$ numbers, the
quaternion skew field $\bf H$ and the octonion algebra $\bf O$. 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 $(f,g)\mapsto f^{-1}g$ 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.
