  This note provides a succinct survey of the existing literature concerning
the H\"older regularity for the gradient of weak solutions of PDEs of the form
$$\sum_{i=1}^{2n} X_i A_i(\nabla_0 u)=0 \text{ and } \partial_t u=
\sum_{i=1}^{2n} X_i A_i(\nabla_0 u)$$ modeled on the $p$-Laplacian in a domain
$\Omega$ in the Heisenberg group $\mathbb H^n$, with $1\le p <\infty$, and of
its parabolic counterpart. We present some open problems and outline some of
the difficulties they present.
