Features of quantum geometry will be illustrated through examples. Specifically, we will consider: i) Einstein-Maxwell theory in 2+1 dimensions; and, ii) quantum geometry of black hole horizons in 3+1 dimensions. In the first case, one can completely solve the model, write the quantum metric operator and analyze its properties. The analysis brings out some unforeseen limitations of the classical and semi-classical theory. In the second case, the horizon geometry is captured by the quantum Chern-Simons theory on a punctured 2-sphere. These states account for the black hole entropy.  The horizon resembles a ``pinned balloon''. At each puncture there is an effective deficit angle and all these angles add up to 4?.  Both treatments are non-perturbative and quantum effects are associated with "Coulmobic'' rather than "radiative'' modes of the gravitational field.