  We study the global topological structure and smoothness of the boundaries of
$\varepsilon$-neighbourhoods $E_\varepsilon = \{x \in \mathbb{R}^2 \, : \,
\textrm{dist}(x, E) \leq \varepsilon \}$ of planar sets $E \subset
\mathbb{R}^2$. We show that for a compact set $E$ and $\varepsilon > 0$ the
boundary $\partial E_\varepsilon$ can be expressed as a disjoint union of an at
most countably infinite union of Jordan curves and a possibly uncountable,
totally disconnected set of singularities. We also show that curvature is
defined almost everywhere on the Jordan curve subsets of the boundary.
