  This survey contains an exposition of ideas and results related to Faltings'
proof of the conjectures of Shafarevich, Tate and Mordell.
  This paper originally appeared in 1986 as an Appendix to the Russian
translation of Serge Lang, "Fundamentals of Diophantine Geometry" (Springer
Verlag, 1983) published by "Mir", Moscow (MR0854670, 88a:11054). A history of
the publication of the Appendix is briefly described by Lang in Section 4 of
his paper "Mordell's review, Siegel's letter to Mordell, Diophantine geometry,
and 20th century mathematics" that was published (in 1995) simultaneously in
Notices of the AMS and Gazette des Math\'ematiciens (SMF) (MR1316025,
96g:11002a; MR1316133, 96g:11002b)
http://smf.emath.fr/Publications/Gazette/1995/63/smf_gazette_63_17-36.pdf .
  Later an expanded version of the Appendix was translated into English by Neal
Koblitz and published in 1989 by the American Mathematical Society as part of
the collection "Eight papers translated from the Russian", AMS Translations,
Series 2, Vol. 143 http://www.ams.org/bookstore-getitem/item=TRANS2-143
(MR1008476, 90b:00011).
  We put this paper on the arXiv with the kind permission of the American
Mathematical Society. For this version we slightly updated the bibliography and
added a few short notes (marked as "Added in December 2009"). We also corrected
inaccuracies that were kindly pointed out to us by J.-P. Serre - one of the few
people to read this paper (in English) twenty years ago.

Number Theory;Algebraic Geometry;History and Overview