Non-commutative gauge theories with a non-constant NC-parameter are investigated. As a novel approach, we propose that such theories should admit an underlying $L_{\infty}$ algebra, that governs not only the action of the symmetries but also the dynamics of the theory. Our approach is well motivated from string theory. In this talk I will give a brief introduction to $L_{\infty}$ algebras and discuss in more details the $L_{\infty}$ bootstrap program: the existence of the solution, uniqueness and particular examples. The talk is mainly based on: arXiv:1803.00732 and 1806.10314.