Given a set $K$ in a metric space $M$ one may ask when is $K$ contained in
$\Gamma$, a connected set of finite 1-dimensional Hausdorff length, and
for estimates on the minimal length of such a $\Gamma$ (up to
multiplicative constants). This was first answered for $M$ = the
Euclidean plane by P. Jones and extended to $M=R^d$ by K. Okikiolu.
Recently, there have been several new relevant results (by several people)
for $M$ being: a Hilbert space, the Heisenberg group, and a general metric
space. for some of these one restricts the discussion to $K$ and $\Gamma$
in specific categories. In some of these categories which we will discuss
the anwswer is in IFF form, whereas in others it is not. The answer to
this question usually comes together with a multiresolutional analysis of
the set $K$ and a construciton of a $\Gamma$ containing $K$ which is not
'too long'. Essentially no prior knowledge in analysis is needed to
understand this talk.