ABSTRACT
This chapter is devoted to the study of generalizations to higher codimensions of the concept of isoparametric hypersurface. The beginning of this topic goes back to the 1980s, with the (sometimes independent) work of many authors, in particular Eells [119], Gromoll and Grove [142], Harle [143], Stru¨bing [302], Terng [316], and Wang [341]. There are different aspects of these generalizations that are closely related: isoparametric maps, isoparametric submanifolds, and submanifolds with constant principal curvatures (for more details on the historical development, see [320]).
