  A \emph{surface of translation} is a sum
$(u,v)\mapsto\gt\alpha(u)+\gt\beta(v)$ of two space curves: a \emph{path}
$\gt\alpha$ and a \emph{profile} $\gt\beta$. A fundamental problem of
differential geometry and shell theory is to determine the ways in which
surfaces deform isometrically, i.e., by bending without stretching. Here, we
explore how surfaces of translation bend. Existence conditions and closed-form
expressions for special bendings of the infinitesimal and finite kinds are
provided. In particular, all surfaces of translation admit a purely torsional
infinitesimal bending. Surfaces of translation whose path and profile belong to
an elliptic cone or to two planes but never to their intersection further admit
a torsion-free infinitesimal bending. Should the planes be orthogonal, the
infinitesimal bending can be integrated into a torsion-free (finite) bending.
Surfaces of translation also admit a torsion-free bending if the path or
profile has exactly two tangency directions. Throughout, smooth and piecewise
smooth surfaces, i.e., surfaces with straight or curved creases, are invariably
dealt with and some extra care is given to situations where the bendings cause
new creases to emerge.
