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Low-energy Coulomb excitation provides a well-understood means of exciting atomic nuclei and allows measuring electromagnetic moments that can be directly related to the nuclear shape. The availability of radioactive ion beams (RIBs) at energies near the Coulomb barrier has made it possible to study shape coexistence in a variety of short-lived exotic nuclei. This review presents a short overview of the methods related to multi-step Coulomb excitation experiments, followed by a discussion of several examples. The focus is on two mass regions where recent Coulomb excitation experiments have contributed to the quantitative understanding of shape coexistence: nuclei with mass near the N = Z line and nuclei with near neutron number N = 60. Experimental results are summarized and their significance for understanding shape coexistence is discussed. Experimental observables such as quadrupole moments and electromagnetic transition strengths represent furthermore important benchmarks for advancing theoretical nuclear structure models. With several new RIB facilities planned and under construction, Coulomb excitation will remain to be an important tool to extend the studies of nuclear shapes toward more exotic systems, and to obtain a more comprehensive and quantitative understanding of shape coexistence. [ABSTRACT FROM AUTHOR]