Ordinary higher gauge theory suffers from the problem that all the curvature forms but the top degree one, which are called fake curvatures, have to vanish. If this condition was omitted, the gauge structure, the underlying higher principal bundle and the corresponding parallel transport would be inconsistent. For vanishing fake curvatures, however, one can locally gauge away the non-abelian parts of the higher connection, ending up with a connection on an abelian gerbe. This is clearly unsatisfactory for non-topological higher gauge theories. A solution to this problem is what we call "adjusted higher gauge theory", in which the usual definition of the curvatures is adjusted by additional data. This lifts the requirement for vanishing fake curvatures. Moreover, it matches constructions of theoretical physicists in the context of supergravity. In this talk, I will review the above points and say a few words about my motivation for studying adjusted higher gauge theories.