Hydrogen atoms are usually considered chemisorbed at well-defined sites on surfaces. We advocate a completely different view, and demonstrate that chemisorbed hydrogen exhibits pronounced quantum effects. The hydrogen atom is to a large degree delocalized in both ground and excited-stated configurations: a proper description can only be given in terms of hydrogen energy bands. An analogous picture emerges for hydrogen isotopes (including the muon) diffusing interstitially in bulk metals. The ground state there corresponds to a self-trapped situation: a localized impurity with an associated lattice distortion field. A powerful computational scheme is presented, which entails (i) the construction of the potential energy field by the effective-medium theory; (ii) the three-dimensional solution of the hydrogen distribution from the protonic Schrödinger equation; (iii) the calculation of the forces exerted on the host atoms and their displacements; and (iv) the iteration to self-consistency. Examples are discussed for both bulk and surface hydrogen.