  Submanifolds of finite type were introduced by the author during the late
1970s. The first results on this subject were collected in author's books
[26,29]. In 1991, a list of twelve open problems and three conjectures on
finite type submanifolds was published in [40]. A detailed survey of the
results, up to 1996, on this subject was given by the author in [48]. Recently,
the study of finite type submanifolds, in particular, of biharmonic
submanifolds, have received a growing attention with many progresses since the
beginning of this century. In this article, we provide a detailed account of
recent development on the problems and conjectures listed in [40].
