A Manifestly Gauge-Invariant Approach to Quantum Theories of Gauge Fields

In gauge theories, physical histories are represented by space-time connections modulo gauge transformations. The space of histories is thus intrinsically non-linear. The standard framework of constructive quantum field theory has to be extended to face these {\it kinematical} non-linearities squarely. We first present a pedagogical account of this problem and then suggest an avenue for its resolution.