| Summary: | The algebraic group theory approach to pairing in nuclei is an old subject and yet it continues to be important in nuclear structure, giving new results. It is well known that for identical nucleons in the shell model approach with <i>j</i> − <i>j</i> coupling, pairing algebra is <i>SU</i>(2) with a complementary number-conserving <i>Sp</i>(<i>N</i>) algebra and for nucleons with good isospin, it is <i>SO</i>(5) with a complementary number-conserving <i>Sp</i>(2Ω) algebra. Similarly, with <i>L</i> − <i>S</i> coupling and isospin, the pairing algebra is <i>SO</i>(8). On the other hand, in the interacting boson models of nuclei, with identical bosons (IBM-1) the pairing algebra is <i>SU</i>(1, 1) with a complementary number-conserving <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>O</mi><mo stretchy="false">(</mo><mi mathvariant="script">N</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> algebra and for the proton–neutron interacting boson model (IBM-2) with good <i>F</i>-spin, it is <i>SO</i>(3, 2) with a complementary number-conserving <i>SO</i>(Ω<i><sup>B</sup></i>) algebra. Furthermore, in IBM-3 and IBM-4 models several pairing algebras are possible. With more than one <i>j</i> or <i>ℓ</i> orbit in shell model, i.e., in the multi-orbit situation, the pairing algebras are not unique and we have the new <i>paradigm</i> of multiple pairing [<i>SU</i>(2), <i>SO</i>(5) and <i>SO</i>(8)] algebras in shell models and similarly there are multiple pairing algebras [<i>SU</i>(1, 1), <i>SO</i>(3, 2) etc.] in interacting boson models. A review of the results for multiple multi-orbit pairing algebras in shell models and interacting boson models is presented in this article with details given for multiple <i>SU</i>(2), <i>SO</i>(5), <i>SU</i>(1, 1) and <i>SO</i>(3, 2) pairing algebras. Some applications of these multiple pairing algebras are discussed. Finally, multiple <i>SO</i>(8) pairing algebras in shell model and pairing algebras in IBM-3 model are briefly discussed.
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