The nucleus is a unique laboratory in physics — a quantum many-body system comprising two types of fermion, the neutron and proton, differing in charge but otherwise essentially identical in their behaviour. The fact that the strong interaction between these fermions is largely independent of charge results in striking symmetries in nuclei. This neutron–proton exchange invariance is encompassed in the elegant concept and formalism of Wigner's isotopic spin — or isospin. The impact of isospin symmetry is maximal near the N=Z line where nuclei have equal numbers of neutrons and protons, and studies involving isospin effects have undergone a resurgence in recent years as such nuclei become more readily accessible. In this review we discuss three isospin-related phenomena: the elegant isospin symmetry of excited analogue states in nuclei, the origin of the extra binding for nuclei with equal numbers of neutrons and protons and the exotic phenomenon of neutron–proton pairing. These three topics, all of considerable current interest, demonstrate the power, simplicity and modern relevance of the isospin concept.