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 absorb 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 multiwavelength techniques that are currently being employed to identify obscured AGN, and we 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 we highlight some of the key outstanding questions.