Gauge Theory on Twisted $\kappa$-Minkowski: Old Problems and Possible Solutions

We review the application of twist deformation formalism and the construction of noncommutative gauge theory on $\kappa$-Minkowski space-time. We compare two different types of twists: the Abelian and the Jordanian one. In each case we provide the twisted differential calculus and consider $U(1)$ gauge theory. Different methods of obtaining a gauge invariant action and related problems are thoroughly discussed.