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This is the fourth of a series of papers surveying some small part of theremarkable work of our friend and colleague Nigel Kalton. We have written it aspart of a tribute to his memory. It contains almost no new results. This timewe discuss Nigel's partial solutions (obtained jointly with one of us) of theproblem of whether the complex method of interpolation preserves thecompactness of operators. This problem is now 51 years old and still lacks acomplete solution. We also survey some other partial solutions of this problem,obtained before and after the above mentioned joint work. We are simultaneouslyposting a preliminary version of a technical sequel to this paper, whichcontains some small new results. Its future more elaborate version willprobably conclude this series devoted to Nigel's research.

Nigel Kalton and complex interpolation of compact operators