  Starting with the work of Preiss on the geometry of measures, the
classification of uniform measures in $\mathbb R^d$ has remained open, except
for $d=1$ and for compactly supported measures in $d=2$, and for codimension
$1$. In this paper we study $1$-dimensional measures in $\mathbb R^d$ for all
$d$ and classify uniform measures with connected $1$-dimensional support, which
turn out to be homogeneous measures. We provide as well a partial
classification of general uniform measures of dimension $1$ in the absence of
the connected support hypothesis.
