Let $\mathcal{M}$ be a manifold and $f$ be a vector field on $\mathcal{M}$. Let $\exp f = \phi(1)$ where $\phi$ is defined by \begin{align*} \phi(0) &= \mathrm{id}_\mathcal{M} \\ \dot{\phi}(t) &= f \circ \phi(t) \end{align*}
Under what conditions is $\exp f$ a $C^r$ diffeomorphism? What are the smoothness constraints that need to be imposed on $f$? Conversely, if $f$ is $C^r$, what can we say about $\exp f$?