A survey is given of recent developments in the theory of collective vibrations in nuclei based on the shell model. This covers work up to and including the International Conference on Nuclear Physics in Paris during 1964. The emphasis is on the application of the theory, which is illustrated by considering specific nuclei, for example leO, 40Ca and zosPb, for the simplest case of nuclei having both the neutron and proton shells closed. The nickel, tin and lead regions are studied to show the effect of pairing, when the shells are no longer closed but the nuclei are still assumed to be spherical. The additional complications with deformed nuclei are illustrated in the rare earth and transuranic regions, and also with magnesium.

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