The material surrounding accreting supermassive black holes connects the active galactic nucleus with its host galaxy and, besides being responsible for feeding the black hole, provides important information on the feedback that nuclear activity produces on the galaxy. In this Review, we summarize our current understanding of the close environment of accreting supermassive black holes obtained from studies of local active galactic nuclei carried out in the infrared and X-ray regimes. The structure of this circumnuclear material is complex, clumpy and dynamic, and its covering factor depends on the accretion properties of the active galactic nucleus. In the infrared, this obscuring material is a transition zone between the broad- and narrow-line regions, and, at least in some galaxies, it consists of two structures: an equatorial disk/torus and a polar component. In the X-ray regime, the obscuration is produced by multiple absorbers across various spatial scales, mostly associated with the torus and the broad-line region. In the coming decade, the new generation of infrared and X-ray facilities will greatly contribute to our understanding of the structure and physical properties of nuclear obscuration in active galactic nuclei.