  We develop an explicit theory of Kummer varieties associated to Jacobians of
hyperelliptic curves of genus 3, over any field $k$ of characteristic $\neq 2$.
In particular, we provide explicit equations defining the Kummer variety
$\mathcal K$ as a subvariety of $\mathbb P^7$, together with explicit
polynomials giving the duplication map on $\mathcal K$. A careful study of the
degenerations of this map then forms the basis for the development of an
explicit theory of heights on such Jacobians when $k$ is a number field. We use
this input to obtain a good bound on the difference between naive and canonical
height, which is a necessary ingredient for the explicit determination of the
Mordell-Weil group. We illustrate our results with two examples.
