  We survey several mathematical developments in the holonomy approach to gauge
theory. A cornerstone of this approach is the introduction of group structures
on spaces of based loops on a smooth manifold, relying on certain homotopy
equivalence relations -- such as the so-called thin homotopy -- and the
resulting interpretation of gauge fields as group homomorphisms to a Lie group
$G$ satisfying a suitable smoothness condition, encoding the holonomy of a
gauge orbit of smooth connections on a principal $G$-bundle. We also prove
several structural results on thin homotopy, and in particular we clarify the
difference between thin equivalence and retrace equivalence for
piecewise-smooth based loops on a smooth manifold, which are often used
interchangeably in the physics literature. We conclude by listing a set of
questions on topological and functional analytic aspects of groups of based
loops, which we consider to be fundamental to establish a rigorous differential
geometric foundation of the holonomy formulation of gauge theory.
