We consider the inverse connection problem consisting of determining a gauge field on $\mathbb{R}^d$ from its non-abelian Radon
transform along oriented straight lines. The determination is considered modulo gauge transformations. Our results
include: global uniqueness theorems for $d\geq3$, new local uniqueness theorems for $d=2$, constructive proofs
(i.e. proofs containing reconstruction procedures), counterexamples to the global uniqueness for $d=2$, a reduction to
the attenuated X-ray transform.
AMS 2000 Mathematics subject classification: Primary 53C65; 81U40
