By a torsion of a general connection $\Gamma$ on a fibered manifold $Y\to M$ we understand the Frölicher-Nijenhuis bracket of $\Gamma$ and some canonical tangent valued one-form (affinor) on $Y$. Using all natural affinors on higher order cotangent bundles, we determine all torsions of general connections on such bundles. We present the geometrical interpretation and study some properties of the torsions.

We introduce the concept of a dynamical connection on a time-dependent Weil bundle and we characterize the structure of dynamical connections. Then we describe all torsions of dynamical connections.

A total connection of order $r$ in a Lie groupoid $\Phi$ over $M$ is defined as a first order connections in the $(r-1)$-st jet prolongations of $\Phi$. A connection in the groupoid $\Phi$ together with a linear connection on its base, ie. in the groupoid $\Pi \left(M\right)$, give rise to a total connection of order $r$, which is called simple. It is shown that this simple connection is curvature-free iff the generating connections are. Also, an $r$-th order total connection in $\Phi$ defines a total reduction of the $r$-th prolongation of $\Phi$ to $\Phi \times \Pi \left(M\right)$....