  We develop honest and locally adaptive confidence bands for probability
densities. They provide substantially improved confidence statements in case of
inhomogeneous smoothness, and are easily implemented and visualized. The
article contributes conceptual work on locally adaptive inference as a
straightforward modification of the global setting imposes severe obstacles for
statistical purposes. Among others, we introduce a statistical notion of local
H\"older regularity and prove a correspondingly strong version of local
adaptivity. We substantially relax the straightforward localization of the
self-similarity condition in order not to rule out prototypical densities. The
set of densities permanently excluded from the consideration is shown to be
pathological in a mathematically rigorous sense. On a technical level, the
crucial component for the verification of honesty is the identification of an
asymptotically least favorable stationary case by means of Slepian's comparison
inequality.
