  In this article we investigate the structure of uniformly $k$-connected and
uniformly $k$-edge-connected graphs. Whereas both types have previously been
studied independent of each other, we analyze relations between these two
classes. We prove that any uniformly $k$-connected graph is also uniformly
$k$-edge-connected for $k\le 3$ and demonstrate that this is not the case for
$k>3$. Furthermore, uniformly $k$-connected and uniformly $k$-edge-connected
graphs are well understood for $k\le 2$ and it is known how to construct
uniformly $3$-edge-connected graphs. We contribute here a constructive
characterization of uniformly $3$-connected graphs that is inspired by Tuttes
Wheel Theorem. Eventually, these results help us to prove a tight bound on the
number of vertices of minimum degree in uniformly $3$-connected graphs.
