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In a 1966 lecture at Sussex, Hugo Steinhaus (author of
Mathematical Snapshots) conjectured that every smooth loop in
$R^3$ has a pair of parallel tangent lines. The very next
day, Beniamino Segre responded with a lecture claiming to
construct a loop having no such pair---what M. Ghomi and I
term a ${\bf skew loop}$. Segre's construction had gaps, but he
eventually published a correct version, together with many
other interesting results, such as: $\S^2$ admits no
skew loops. I will discuss Segre's work and explain recent
extensions by Ghomi and myself. Studying the relationship
between skew loops and surfaces, for instance, we prove a
strong converse to Segre's result which implies in particular
that among all compact surfaces in  $R^3$,
ellipsoids admit no skew loops.
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