Shape coexistence in odd-mass nuclei near closed shells (±1 and ±3 nucleons) is reviewed. Two main approaches are presented. The more historical one starts from the explicit introduction of a deformation degree of freedom and allows particular Nilsson orbitals to occur very low in excitation energy. A second approach starts from spherical particle-hole (p-h) excitations across a closed shell and allows interactions with quadrupole vibrations of the underlying core nucleus. The equivalence between the two approaches is discussed in some detail. Attention is given also to other approaches to the description of coexistence. A detailed review of the experimental evidence for coexistence in odd-mass nuclei near the Z = 28, 50, 82 and N = 28, 50, 82 closed shells is made. In addition, the fingerprints of coexisting states, i.e. the most pronounced nuclear structure properties characterizing such states, are presented. These fingerprints provide a necessary (though not sufficient) set of conditions for identifying coexistence throughout the nuclear mass table. At the end, we briefly describe the connection of the present study to other areas of nuclear structure such as related configurations in doubly-even nuclei, many-particle many-hole excitations, deep-lying hole states, and the possibility of incorporating these degrees of freedom into an Interacting Boson Approximation (IBA) framework. A major conclusion of this review is that there is a clear connection between shape coexistence and shell-model intruder states. A variety of topics is suggested for future experimental and theoretical investigation.