  We consider random walks and L\'evy processes in a homogeneous group $G$. For
all $p > 0$, we completely characterise (almost) all $G$-valued L\'evy
processes whose sample paths have finite $p$-variation, and give sufficient
conditions under which a sequence of $G$-valued random walks converges in law
to a L\'evy process in $p$-variation topology. In the case that $G$ is the free
nilpotent Lie group over $\mathbb{R}^d$, so that processes of finite
$p$-variation are identified with rough paths, we demonstrate applications of
our results to weak convergence of stochastic flows and provide a
L\'evy-Khintchine formula for the characteristic function of the signature of a
L\'evy process. At the heart of our analysis is a criterion for tightness of
$p$-variation for a collection of c\`adl\`ag strong Markov processes.
