  Active Galactic Nuclei (AGN) are powered by the accretion of material onto a
supermassive black hole (SMBH), and are among the most luminous objects in the
Universe. However, the huge radiative power of most AGN cannot be seen
directly, as the accretion is "hidden" behind gas and dust that absorbs many of
the characteristic observational signatures. This obscuration presents an
important challenge for uncovering the complete AGN population and
understanding the cosmic evolution of SMBHs. In this review we describe a broad
range of multi-wavelength techniques that are currently employed to identify
obscured AGN, and assess the reliability and completeness of each technique. We
follow with a discussion of the demographics of obscured AGN activity, explore
the nature and physical scales of the obscuring material, and assess the
implications of obscured AGN for observational cosmology. We conclude with an
outline of the prospects for future progress from both observations and
theoretical models, and highlight some of the key outstanding questions.
