We give a brief overview of the history, state of the art, and future for
elliptical superconducting cavities. Principles of the cell shape optimization,
criteria for multi-cell structures design, HOM damping schemes and other
features are discussed along with examples of superconducting structures for
various applications.Comment: 25 pages, contribution to the CAS - CERN Accelerator School:
Specialised Course on RF for Accelerators; 8 - 17 Jun 2010, Ebeltoft, Denmar