Experimental and theoretical research on the electronic structure of crystal surfaces over the past 10--15 years is reviewed. The nature of the Tamm (surface) levels of metals and semiconductors, the conditions for their existence, and their spectrum is discussed. Emphasis is placed on the role played by surface states in the reconstruction of surfaces and of the atomic structure and on how surface states affect the surface properties of magnets and work functions. Certain aspects of the formation of the chemisorption bond are discussed briefly.