In their seminal article published in 1998, Donaldson and Thomas proposed the study of gauge theory in higher dimensions, which has seen significant interest in recent years. This theory requires manifolds of dimensions 6, 7 and 8 endowed with geometric structures that induce Riemannian metrics with special holonomy. In this minicourse I will first describe the basics of these manifolds with special holonomy and gauge theory in this setting, before detailing some of the key questions and challenges in the field. I will also discuss a range of the results which have been obtained in gauge theory and special holonomy to give a flavour for recent progress and open problems.